Find the Local Maxima and Minima f(x)=x^4+x^3-6x^2

$f (x) = x^{4} + x^{3} - 6 x^{2}$

Find the first derivative of the function.

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Differentiate.

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By the Sum Rule, the derivative of $x^{4} + x^{3} - 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 x^{3} + 3 x^{2} + \frac{d}{d x} [- 6 x^{2}]$

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} + 3 x^{2} - 6 \frac{d}{d x} [x^{2}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$4 x^{3} + 3 x^{2} - 6 (2 x)$

Multiply $2$ by $- 6$ .

$4 x^{3} + 3 x^{2} - 12 x$

Find the second derivative of the function.

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By the Sum Rule, the derivative of $4 x^{3} + 3 x^{2} - 12 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 12 x]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 12 x)$

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 12 x)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 12 x)$

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 12 x)$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} + 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 12 x)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 12 x^{2} + 3 (2 x) + \frac{d}{d x} (- 12 x)$

Multiply $2$ by $3$ .

$f'' (x) = 12 x^{2} + 6 x + \frac{d}{d x} (- 12 x)$

Evaluate $\frac{d}{d x} [- 12 x]$ .

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Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x$ with respect to $x$ is $- 12 \frac{d}{d x} [x]$ .

$f'' (x) = 12 x^{2} + 6 x - 12 \frac{d}{d x} x$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 12 x^{2} + 6 x - 12 \cdot 1$

Multiply $- 12$ by $1$ .

$f'' (x) = 12 x^{2} + 6 x - 12$

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$4 x^{3} + 3 x^{2} - 12 x = 0$

Factor $x$ out of $4 x^{3} + 3 x^{2} - 12 x$ .

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Factor $x$ out of $4 x^{3}$ .

$x (4 x^{2}) + 3 x^{2} - 12 x = 0$

Factor $x$ out of $3 x^{2}$ .

$x (4 x^{2}) + x (3 x) - 12 x = 0$

Factor $x$ out of $- 12 x$ .

$x (4 x^{2}) + x (3 x) + x \cdot - 12 = 0$

Factor $x$ out of $x (4 x^{2}) + x (3 x)$ .

$x (4 x^{2} + 3 x) + x \cdot - 12 = 0$

Factor $x$ out of $x (4 x^{2} + 3 x) + x \cdot - 12$ .

$x (4 x^{2} + 3 x - 12) = 0$

Set $x$ equal to $0$ and solve for $x$ .

$x = 0$

Set $4 x^{2} + 3 x - 12$ equal to $0$ and solve for $x$ .

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Set the factor equal to $0$ .

$4 x^{2} + 3 x - 12 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 4$ , $b = 3$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (4 \cdot - 12)}}{2 \cdot 4}$

Simplify.

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (4 \cdot - 12)}}{2 \cdot 4}$

Multiply $4$ by $- 12$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 48}}{2 \cdot 4}$

Multiply $- 4$ by $- 48$ .

$x = \frac{- 3 \pm \sqrt{9 + 192}}{2 \cdot 4}$

Add $9$ and $192$ .

$x = \frac{- 3 \pm \sqrt{201}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{201}}{8}$

Factor $- 1$ out of $- 3 \pm \sqrt{201}$ .

$x = \frac{- 13 \pm \sqrt{201}}{8}$

Multiply $- 1$ by $- 1$ .

$x = \frac{1 - 3 \pm \sqrt{201}}{8}$

Multiply $- 3 \pm \sqrt{201}$ by $1$ .

$x = \frac{- 3 \pm \sqrt{201}}{8}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (4 \cdot - 12)}}{2 \cdot 4}$

Multiply $4$ by $- 12$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 48}}{2 \cdot 4}$

Multiply $- 4$ by $- 48$ .

$x = \frac{- 3 \pm \sqrt{9 + 192}}{2 \cdot 4}$

Add $9$ and $192$ .

$x = \frac{- 3 \pm \sqrt{201}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{201}}{8}$

Factor $- 1$ out of $- 3 \pm \sqrt{201}$ .

$x = \frac{- 13 \pm \sqrt{201}}{8}$

Multiply $- 1$ by $- 1$ .

$x = \frac{1 - 3 \pm \sqrt{201}}{8}$

Multiply $- 3 \pm \sqrt{201}$ by $1$ .

$x = \frac{- 3 \pm \sqrt{201}}{8}$

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{201}}{8}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{201}}{8}$

Factor $- 1$ out of $\sqrt{201}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{201})}{8}$

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{201})$ .

$x = \frac{- 1 (3 - \sqrt{201})}{8}$

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{201}}{8}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot (4 \cdot - 12)}}{2 \cdot 4}$

Multiply $4$ by $- 12$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 48}}{2 \cdot 4}$

Multiply $- 4$ by $- 48$ .

$x = \frac{- 3 \pm \sqrt{9 + 192}}{2 \cdot 4}$

Add $9$ and $192$ .

$x = \frac{- 3 \pm \sqrt{201}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{201}}{8}$

Factor $- 1$ out of $- 3 \pm \sqrt{201}$ .

$x = \frac{- 13 \pm \sqrt{201}}{8}$

Multiply $- 1$ by $- 1$ .

$x = \frac{1 - 3 \pm \sqrt{201}}{8}$

Multiply $- 3 \pm \sqrt{201}$ by $1$ .

$x = \frac{- 3 \pm \sqrt{201}}{8}$

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{201}}{8}$

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{201}}{8}$

Factor $- 1$ out of $- \sqrt{201}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{201})}{8}$

Factor $- 1$ out of $- 1 (3) - (\sqrt{201})$ .

$x = \frac{- 1 (3 + \sqrt{201})}{8}$

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{201}}{8}$

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{201}}{8}, - \frac{3 + \sqrt{201}}{8}$

The solution is the result of $x = 0$ and $4 x^{2} + 3 x - 12 = 0$ .

$x = 0, - \frac{3 - \sqrt{201}}{8}, - \frac{3 + \sqrt{201}}{8}$

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(0)}^{2} + 6 (0) - 12$

Evaluate the second derivative.

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Simplify each term.

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Raising $0$ to any positive power yields $0$ .

$12 \cdot 0 + 6 (0) - 12$

Multiply $12$ by $0$ .

$0 + 6 (0) - 12$

Multiply $6$ by $0$ .

$0 + 0 - 12$

Simplify by adding zeros.

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Add $0$ and $0$ .

$0 - 12$

Subtract $12$ from $0$ .

$- 12$

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Find the y-value when $x = 0$ .

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Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{4} + {(0)}^{3} - 6 {(0)}^{2}$

Simplify the result.

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Simplify each term.

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Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + {(0)}^{3} - 6 {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 - 6 {(0)}^{2}$

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 - 6 \cdot 0$

Multiply $- 6$ by $0$ .

$f (0) = 0 + 0 + 0$

Simplify by adding zeros.

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Add $0$ and $0$ .

$f (0) = 0 + 0$

Add $0$ and $0$ .

$f (0) = 0$

The final answer is $0$ .

$y = 0$

Evaluate the second derivative at $x = - \frac{3 - \sqrt{201}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(- \frac{3 - \sqrt{201}}{8})}^{2} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Evaluate the second derivative.

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 - \sqrt{201}}{8}$ .

$12 ({(- 1)}^{2} {(\frac{3 - \sqrt{201}}{8})}^{2}) + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Apply the product rule to $\frac{3 - \sqrt{201}}{8}$ .

$12 ({(- 1)}^{2} \frac{{(3 - \sqrt{201})}^{2}}{8^{2}}) + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Raise $- 1$ to the power of $2$ .

$12 (1 \frac{{(3 - \sqrt{201})}^{2}}{8^{2}}) + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Multiply $\frac{{(3 - \sqrt{201})}^{2}}{8^{2}}$ by $1$ .

$12 \frac{{(3 - \sqrt{201})}^{2}}{8^{2}} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Raise $8$ to the power of $2$ .

$12 \frac{{(3 - \sqrt{201})}^{2}}{64} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Cancel the common factor of $4$ .

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Factor $4$ out of $12$ .

$4 (3) \frac{{(3 - \sqrt{201})}^{2}}{64} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Factor $4$ out of $64$ .

$4 \cdot 3 \frac{{(3 - \sqrt{201})}^{2}}{4 \cdot 16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Cancel the common factor.

$4 \cdot 3 \frac{{(3 - \sqrt{201})}^{2}}{4 \cdot 16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Rewrite the expression.

$3 \frac{{(3 - \sqrt{201})}^{2}}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Combine $3$ and $\frac{{(3 - \sqrt{201})}^{2}}{16}$ .

$\frac{3 {(3 - \sqrt{201})}^{2}}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Rewrite ${(3 - \sqrt{201})}^{2}$ as $(3 - \sqrt{201}) (3 - \sqrt{201})$ .

$\frac{3 ((3 - \sqrt{201}) (3 - \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Expand $(3 - \sqrt{201}) (3 - \sqrt{201})$ using the FOIL Method.

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Apply the distributive property.

$\frac{3 (3 (3 - \sqrt{201}) - \sqrt{201} (3 - \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 (- \sqrt{201}) - \sqrt{201} (3 - \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 (- \sqrt{201}) - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $3$ .

$\frac{3 (9 + 3 (- \sqrt{201}) - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Multiply $- 1$ by $3$ .

$\frac{3 (9 - 3 \sqrt{201} - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Multiply $3$ by $- 1$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} - \sqrt{201} (- \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Multiply $- \sqrt{201} (- \sqrt{201})$ .

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Multiply $- 1$ by $- 1$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 1 \sqrt{201} \sqrt{201})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Multiply $\sqrt{201}$ by $1$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + \sqrt{201} \sqrt{201})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Raise $\sqrt{201}$ to the power of $1$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{1} \sqrt{201})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Raise $\sqrt{201}$ to the power of $1$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{1} {\sqrt{201}}^{1})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{1 + 1})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Add $1$ and $1$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{2})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {(201^{\frac{1}{2}})}^{2})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{1}{2} \cdot 2})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{2}{2}})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{2}{2}})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Divide $1$ by $1$ .

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{1})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Evaluate the exponent.

$\frac{3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201)}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Add $9$ and $201$ .

$\frac{3 (210 - 3 \sqrt{201} - 3 \sqrt{201})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Subtract $3 \sqrt{201}$ from $- 3 \sqrt{201}$ .

$\frac{3 (210 - 6 \sqrt{201})}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Cancel the common factor of $210 - 6 \sqrt{201}$ and $16$ .

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Factor $2$ out of $3 (210 - 6 \sqrt{201})$ .

$\frac{2 (3 (105 - 3 \sqrt{201}))}{16} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Cancel the common factors.

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Factor $2$ out of $16$ .

$\frac{2 (3 (105 - 3 \sqrt{201}))}{2 (8)} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Cancel the common factor.

$\frac{2 (3 (105 - 3 \sqrt{201}))}{2 \cdot 8} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Rewrite the expression.

$\frac{3 (105 - 3 \sqrt{201})}{8} + 6 (- \frac{3 - \sqrt{201}}{8}) - 12$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{3 - \sqrt{201}}{8}$ into the numerator.

$\frac{3 (105 - 3 \sqrt{201})}{8} + 6 \frac{- (3 - \sqrt{201})}{8} - 12$

Factor $2$ out of $6$ .

$\frac{3 (105 - 3 \sqrt{201})}{8} + 2 (3) \frac{- (3 - \sqrt{201})}{8} - 12$

Factor $2$ out of $8$ .

$\frac{3 (105 - 3 \sqrt{201})}{8} + 2 \cdot 3 \frac{- (3 - \sqrt{201})}{2 \cdot 4} - 12$

Cancel the common factor.

$\frac{3 (105 - 3 \sqrt{201})}{8} + 2 \cdot 3 \frac{- (3 - \sqrt{201})}{2 \cdot 4} - 12$

Rewrite the expression.

$\frac{3 (105 - 3 \sqrt{201})}{8} + 3 \frac{- (3 - \sqrt{201})}{4} - 12$

Combine $3$ and $\frac{- (3 - \sqrt{201})}{4}$ .

$\frac{3 (105 - 3 \sqrt{201})}{8} + \frac{3 (- (3 - \sqrt{201}))}{4} - 12$

Multiply $- 1$ by $3$ .

$\frac{3 (105 - 3 \sqrt{201})}{8} + \frac{- 3 (3 - \sqrt{201})}{4} - 12$

Move the negative in front of the fraction.

$\frac{3 (105 - 3 \sqrt{201})}{8} - \frac{3 (3 - \sqrt{201})}{4} - 12$

To write $- \frac{3 (3 - \sqrt{201})}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (105 - 3 \sqrt{201})}{8} - \frac{3 (3 - \sqrt{201})}{4} \cdot \frac{2}{2} - 12$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 (3 - \sqrt{201})}{4}$ and $\frac{2}{2}$ .

$\frac{3 (105 - 3 \sqrt{201})}{8} - \frac{3 (3 - \sqrt{201}) \cdot 2}{4 \cdot 2} - 12$

Multiply $4$ by $2$ .

$\frac{3 (105 - 3 \sqrt{201})}{8} - \frac{3 (3 - \sqrt{201}) \cdot 2}{8} - 12$

Combine the numerators over the common denominator.

$\frac{3 (105 - 3 \sqrt{201}) - 3 (3 - \sqrt{201}) \cdot 2}{8} - 12$

Simplify the numerator.

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Apply the distributive property.

$\frac{3 \cdot 105 + 3 (- 3 \sqrt{201}) - 3 (3 - \sqrt{201}) \cdot 2}{8} - 12$

Multiply $3$ by $105$ .

$\frac{315 + 3 (- 3 \sqrt{201}) - 3 (3 - \sqrt{201}) \cdot 2}{8} - 12$

Multiply $- 3$ by $3$ .

$\frac{315 - 9 \sqrt{201} - 3 (3 - \sqrt{201}) \cdot 2}{8} - 12$

Apply the distributive property.

$\frac{315 - 9 \sqrt{201} + (- 3 \cdot 3 - 3 (- \sqrt{201})) \cdot 2}{8} - 12$

Multiply $- 3$ by $3$ .

$\frac{315 - 9 \sqrt{201} + (- 9 - 3 (- \sqrt{201})) \cdot 2}{8} - 12$

Multiply $- 1$ by $- 3$ .

$\frac{315 - 9 \sqrt{201} + (- 9 + 3 \sqrt{201}) \cdot 2}{8} - 12$

Apply the distributive property.

$\frac{315 - 9 \sqrt{201} - 9 \cdot 2 + 3 \sqrt{201} \cdot 2}{8} - 12$

Multiply $- 9$ by $2$ .

$\frac{315 - 9 \sqrt{201} - 18 + 3 \sqrt{201} \cdot 2}{8} - 12$

Multiply $2$ by $3$ .

$\frac{315 - 9 \sqrt{201} - 18 + 6 \sqrt{201}}{8} - 12$

Subtract $18$ from $315$ .

$\frac{297 - 9 \sqrt{201} + 6 \sqrt{201}}{8} - 12$

Add $- 9 \sqrt{201}$ and $6 \sqrt{201}$ .

$\frac{297 - 3 \sqrt{201}}{8} - 12$

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{297 - 3 \sqrt{201}}{8} - 12 \cdot \frac{8}{8}$

Combine fractions.

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Combine $- 12$ and $\frac{8}{8}$ .

$\frac{297 - 3 \sqrt{201}}{8} + \frac{- 12 \cdot 8}{8}$

Combine the numerators over the common denominator.

$\frac{297 - 3 \sqrt{201} - 12 \cdot 8}{8}$

Simplify the numerator.

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Multiply $- 12$ by $8$ .

$\frac{297 - 3 \sqrt{201} - 96}{8}$

Subtract $96$ from $297$ .

$\frac{201 - 3 \sqrt{201}}{8}$

$x = - \frac{3 - \sqrt{201}}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{3 - \sqrt{201}}{8}$ is a local minimum

Find the y-value when $x = - \frac{3 - \sqrt{201}}{8}$ .

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Replace the variable $x$ with $- \frac{3 - \sqrt{201}}{8}$ in the expression.

$f (- \frac{3 - \sqrt{201}}{8}) = {(- \frac{3 - \sqrt{201}}{8})}^{4} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Simplify the result.

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 - \sqrt{201}}{8}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = {(- 1)}^{4} {(\frac{3 - \sqrt{201}}{8})}^{4} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the product rule to $\frac{3 - \sqrt{201}}{8}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = {(- 1)}^{4} (\frac{{(3 - \sqrt{201})}^{4}}{8^{4}}) + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = 1 (\frac{{(3 - \sqrt{201})}^{4}}{8^{4}}) + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $\frac{{(3 - \sqrt{201})}^{4}}{8^{4}}$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{{(3 - \sqrt{201})}^{4}}{8^{4}} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $8$ to the power of $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{{(3 - \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Use the Binomial Theorem.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{3^{4} + 4 \cdot (3^{3} (- \sqrt{201})) + 6 \cdot (3^{2} {(- \sqrt{201})}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Simplify each term.

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Raise $3$ to the power of $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 + 4 \cdot (3^{3} (- \sqrt{201})) + 6 \cdot (3^{2} {(- \sqrt{201})}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $3$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 + 4 \cdot (27 (- \sqrt{201})) + 6 \cdot (3^{2} {(- \sqrt{201})}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $4$ by $27$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 + 108 (- \sqrt{201}) + 6 \cdot (3^{2} {(- \sqrt{201})}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $- 1$ by $108$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 6 \cdot (3^{2} {(- \sqrt{201})}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $3$ to the power of $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 6 \cdot (9 {(- \sqrt{201})}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $6$ by $9$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 {(- \sqrt{201})}^{2} + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the product rule to $- \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 ({(- 1)}^{2} {\sqrt{201}}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 (1 {\sqrt{201}}^{2}) + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply ${\sqrt{201}}^{2}$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 {\sqrt{201}}^{2} + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 {(201^{\frac{1}{2}})}^{2} + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 \cdot 201^{\frac{1}{2} \cdot 2} + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 \cdot 201^{\frac{2}{2}} + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

Divide $1$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 54 \cdot 201 + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Evaluate the exponent.

Multiply $54$ by $201$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 4 \cdot (3 {(- \sqrt{201})}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $4$ by $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the product rule to $- \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 ({(- 1)}^{3} {\sqrt{201}}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 (- {\sqrt{201}}^{3}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{3}$ as ${(201^{3})}^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{201^{3}}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $201$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{8120601}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Factor $40401$ out of $8120601$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{40401 (201)}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite $40401$ as $201^{2}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{201^{2} \cdot 201}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Pull terms out from under the radical.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 (- (201 \sqrt{201})) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $201$ by $- 1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 + 12 (- 201 \sqrt{201}) + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $- 201$ by $12$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {(- \sqrt{201})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the product rule to $- \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {(- 1)}^{4} {\sqrt{201}}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 1 {\sqrt{201}}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply ${\sqrt{201}}^{4}$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{4}$ as $201^{2}$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {(201^{\frac{1}{2}})}^{4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{1}{2} \cdot 4}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Combine $\frac{1}{2}$ and $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{4}{2}}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2}}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factors.

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Factor $2$ out of $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 (1)}}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite the expression.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2}{1}}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Divide $2$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{2}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $201$ to the power of $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 40401}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Add $81$ and $10854$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{10935 - 108 \sqrt{201} - 2412 \sqrt{201} + 40401}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Add $10935$ and $40401$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{51336 - 108 \sqrt{201} - 2412 \sqrt{201}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Subtract $2412 \sqrt{201}$ from $- 108 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{51336 - 2520 \sqrt{201}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor of $51336 - 2520 \sqrt{201}$ and $4096$ .

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Factor $8$ out of $51336$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{8 (6417) - 2520 \sqrt{201}}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Factor $8$ out of $- 2520 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{8 (6417) + 8 (- 315 \sqrt{201})}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Factor $8$ out of $8 (6417) + 8 (- 315 \sqrt{201})$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{8 (6417 - 315 \sqrt{201})}{4096} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factors.

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Factor $8$ out of $4096$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{8 (6417 - 315 \sqrt{201})}{8 \cdot 512} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{8 (6417 - 315 \sqrt{201})}{8 \cdot 512} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite the expression.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} + {(- \frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 - \sqrt{201}}{8}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} + {(- 1)}^{3} {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the product rule to $\frac{3 - \sqrt{201}}{8}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} + {(- 1)}^{3} (\frac{{(3 - \sqrt{201})}^{3}}{8^{3}}) - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{{(3 - \sqrt{201})}^{3}}{8^{3}} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $8$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{{(3 - \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Use the Binomial Theorem.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{3^{3} + 3 \cdot (3^{2} (- \sqrt{201})) + 3 \cdot (3 {(- \sqrt{201})}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Simplify each term.

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Raise $3$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 3 \cdot (3^{2} (- \sqrt{201})) + 3 \cdot (3 {(- \sqrt{201})}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $3$ by $3^{2}$ by adding the exponents.

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Multiply $3$ by $3^{2}$ .

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Raise $3$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 3^{1 + 2} (- \sqrt{201}) + 3 \cdot (3 {(- \sqrt{201})}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Add $1$ and $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 3^{3} (- \sqrt{201}) + 3 \cdot (3 {(- \sqrt{201})}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $3$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 27 (- \sqrt{201}) + 3 \cdot (3 {(- \sqrt{201})}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $- 1$ by $27$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 3 \cdot (3 {(- \sqrt{201})}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $3$ by $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the product rule to $- \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 ({(- 1)}^{2} {\sqrt{201}}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 (1 {\sqrt{201}}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply ${\sqrt{201}}^{2}$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 {\sqrt{201}}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 {(201^{\frac{1}{2}})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201^{\frac{1}{2} \cdot 2} + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Divide $1$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201 + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Evaluate the exponent.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201 + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $9$ by $201$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 + {(- \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Apply the product rule to $- \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 + {(- 1)}^{3} {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{3}$ as ${(201^{3})}^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{201^{3}}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Raise $201$ to the power of $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{8120601}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Factor $40401$ out of $8120601$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{40401 (201)}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite $40401$ as $201^{2}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{201^{2} \cdot 201}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Pull terms out from under the radical.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - (201 \sqrt{201})}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Multiply $201$ by $- 1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - 201 \sqrt{201}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Add $27$ and $1809$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{1836 - 27 \sqrt{201} - 201 \sqrt{201}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Subtract $201 \sqrt{201}$ from $- 27 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{1836 - 228 \sqrt{201}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor of $1836 - 228 \sqrt{201}$ and $512$ .

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Factor $4$ out of $1836$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459) - 228 \sqrt{201}}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Factor $4$ out of $- 228 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459) + 4 (- 57 \sqrt{201})}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Factor $4$ out of $4 (459) + 4 (- 57 \sqrt{201})$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459 - 57 \sqrt{201})}{512} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factors.

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Factor $4$ out of $512$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459 - 57 \sqrt{201})}{4 \cdot 128} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Cancel the common factor.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459 - 57 \sqrt{201})}{4 \cdot 128} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Rewrite the expression.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 {(- \frac{3 - \sqrt{201}}{8})}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 - \sqrt{201}}{8}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 ({(- 1)}^{2} {(\frac{3 - \sqrt{201}}{8})}^{2})$

Apply the product rule to $\frac{3 - \sqrt{201}}{8}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 ({(- 1)}^{2} (\frac{{(3 - \sqrt{201})}^{2}}{8^{2}}))$

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 (1 (\frac{{(3 - \sqrt{201})}^{2}}{8^{2}}))$

Multiply $\frac{{(3 - \sqrt{201})}^{2}}{8^{2}}$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 \frac{{(3 - \sqrt{201})}^{2}}{8^{2}}$

Raise $8$ to the power of $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 \frac{{(3 - \sqrt{201})}^{2}}{64}$

Cancel the common factor of $2$ .

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Factor $2$ out of $- 6$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + 2 (- 3) (\frac{{(3 - \sqrt{201})}^{2}}{64})$

Factor $2$ out of $64$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + 2 \cdot (- 3 \frac{{(3 - \sqrt{201})}^{2}}{2 \cdot 32})$

Cancel the common factor.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + 2 \cdot (- 3 \frac{{(3 - \sqrt{201})}^{2}}{2 \cdot 32})$

Rewrite the expression.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 3 \frac{{(3 - \sqrt{201})}^{2}}{32}$

Combine $- 3$ and $\frac{{(3 - \sqrt{201})}^{2}}{32}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 {(3 - \sqrt{201})}^{2}}{32}$

Rewrite ${(3 - \sqrt{201})}^{2}$ as $(3 - \sqrt{201}) (3 - \sqrt{201})$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 ((3 - \sqrt{201}) (3 - \sqrt{201}))}{32}$

Expand $(3 - \sqrt{201}) (3 - \sqrt{201})$ using the FOIL Method.

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Apply the distributive property.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (3 (3 - \sqrt{201}) - \sqrt{201} (3 - \sqrt{201}))}{32}$

Apply the distributive property.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 (- \sqrt{201}) - \sqrt{201} (3 - \sqrt{201}))}{32}$

Apply the distributive property.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 (- \sqrt{201}) - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{32}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 (- \sqrt{201}) - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{32}$

Multiply $- 1$ by $3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{32}$

Multiply $3$ by $- 1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} - \sqrt{201} (- \sqrt{201}))}{32}$

Multiply $- \sqrt{201} (- \sqrt{201})$ .

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Multiply $- 1$ by $- 1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 1 \sqrt{201} \sqrt{201})}{32}$

Multiply $\sqrt{201}$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + \sqrt{201} \sqrt{201})}{32}$

Raise $\sqrt{201}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + \sqrt{201} \sqrt{201})}{32}$

Raise $\sqrt{201}$ to the power of $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + \sqrt{201} \sqrt{201})}{32}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{1 + 1})}{32}$

Add $1$ and $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{2})}{32}$

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {(201^{\frac{1}{2}})}^{2})}{32}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{1}{2} \cdot 2})}{32}$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{2}{2}})}{32}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{2}{2}})}{32}$

Divide $1$ by $1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201)}{32}$

Evaluate the exponent.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201)}{32}$

Add $9$ and $201$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (210 - 3 \sqrt{201} - 3 \sqrt{201})}{32}$

Subtract $3 \sqrt{201}$ from $- 3 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (210 - 6 \sqrt{201})}{32}$

Cancel the common factor of $210 - 6 \sqrt{201}$ and $32$ .

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Factor $2$ out of $- 3 (210 - 6 \sqrt{201})$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 - 3 \sqrt{201}))}{32}$

Cancel the common factors.

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Factor $2$ out of $32$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 - 3 \sqrt{201}))}{2 (16)}$

Cancel the common factor.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 - 3 \sqrt{201}))}{2 \cdot 16}$

Rewrite the expression.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (105 - 3 \sqrt{201})}{16}$

Move the negative in front of the fraction.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - \frac{3 (105 - 3 \sqrt{201})}{16}$

To write $- \frac{459 - 57 \sqrt{201}}{128}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} \cdot \frac{4}{4} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Write each expression with a common denominator of $512$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{459 - 57 \sqrt{201}}{128}$ and $\frac{4}{4}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Multiply $128$ by $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Combine the numerators over the common denominator.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201} - (459 - 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Simplify the numerator.

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Apply the distributive property.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201} + (- 1 \cdot 459 - (- 57 \sqrt{201})) \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Multiply $- 1$ by $459$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201} + (- 459 - (- 57 \sqrt{201})) \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Multiply $- 57$ by $- 1$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201} + (- 459 + 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Apply the distributive property.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201} - 459 \cdot 4 + 57 \sqrt{201} \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Multiply $- 459$ by $4$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201} - 1836 + 57 \sqrt{201} \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Multiply $4$ by $57$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201}}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Subtract $1836$ from $6417$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 315 \sqrt{201} + 228 \sqrt{201}}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Add $- 315 \sqrt{201}$ and $228 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201}}{512} - \frac{3 (105 - 3 \sqrt{201})}{16}$

To write $- \frac{3 (105 - 3 \sqrt{201})}{16}$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201}}{512} - \frac{3 (105 - 3 \sqrt{201})}{16} \cdot \frac{32}{32}$

Write each expression with a common denominator of $512$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 (105 - 3 \sqrt{201})}{16}$ and $\frac{32}{32}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201}}{512} - \frac{3 (105 - 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Multiply $16$ by $32$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201}}{512} - \frac{3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Combine the numerators over the common denominator.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201} - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Simplify the numerator.

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Apply the distributive property.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201} + (- 3 \cdot 105 - 3 (- 3 \sqrt{201})) \cdot 32}{512}$

Multiply $- 3$ by $105$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201} + (- 315 - 3 (- 3 \sqrt{201})) \cdot 32}{512}$

Multiply $- 3$ by $- 3$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201} + (- 315 + 9 \sqrt{201}) \cdot 32}{512}$

Apply the distributive property.

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201} - 315 \cdot 32 + 9 \sqrt{201} \cdot 32}{512}$

Multiply $- 315$ by $32$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201} - 10080 + 9 \sqrt{201} \cdot 32}{512}$

Multiply $32$ by $9$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{4581 - 87 \sqrt{201} - 10080 + 288 \sqrt{201}}{512}$

Subtract $10080$ from $4581$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{- 5499 - 87 \sqrt{201} + 288 \sqrt{201}}{512}$

Add $- 87 \sqrt{201}$ and $288 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{- 5499 + 201 \sqrt{201}}{512}$

Simplify with factoring out.

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Rewrite $- 5499$ as $- 1 (5499)$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{- 1 \cdot 5499 + 201 \sqrt{201}}{512}$

Factor $- 1$ out of $201 \sqrt{201}$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{- 1 \cdot 5499 - (- 201 \sqrt{201})}{512}$

Factor $- 1$ out of $- 1 (5499) - (- 201 \sqrt{201})$ .

$f (- \frac{3 - \sqrt{201}}{8}) = \frac{- 1 (5499 - 201 \sqrt{201})}{512}$

Move the negative in front of the fraction.

$f (- \frac{3 - \sqrt{201}}{8}) = - \frac{5499 - 201 \sqrt{201}}{512}$

The final answer is $- \frac{5499 - 201 \sqrt{201}}{512}$ .

$y = - \frac{5499 - 201 \sqrt{201}}{512}$

Evaluate the second derivative at $x = - \frac{3 + \sqrt{201}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$12 {(- \frac{3 + \sqrt{201}}{8})}^{2} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Evaluate the second derivative.

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 + \sqrt{201}}{8}$ .

$12 ({(- 1)}^{2} {(\frac{3 + \sqrt{201}}{8})}^{2}) + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Apply the product rule to $\frac{3 + \sqrt{201}}{8}$ .

$12 ({(- 1)}^{2} \frac{{(3 + \sqrt{201})}^{2}}{8^{2}}) + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Raise $- 1$ to the power of $2$ .

$12 (1 \frac{{(3 + \sqrt{201})}^{2}}{8^{2}}) + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Multiply $\frac{{(3 + \sqrt{201})}^{2}}{8^{2}}$ by $1$ .

$12 \frac{{(3 + \sqrt{201})}^{2}}{8^{2}} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Raise $8$ to the power of $2$ .

$12 \frac{{(3 + \sqrt{201})}^{2}}{64} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Cancel the common factor of $4$ .

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Factor $4$ out of $12$ .

$4 (3) \frac{{(3 + \sqrt{201})}^{2}}{64} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Factor $4$ out of $64$ .

$4 \cdot 3 \frac{{(3 + \sqrt{201})}^{2}}{4 \cdot 16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Cancel the common factor.

$4 \cdot 3 \frac{{(3 + \sqrt{201})}^{2}}{4 \cdot 16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Rewrite the expression.

$3 \frac{{(3 + \sqrt{201})}^{2}}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Combine $3$ and $\frac{{(3 + \sqrt{201})}^{2}}{16}$ .

$\frac{3 {(3 + \sqrt{201})}^{2}}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Rewrite ${(3 + \sqrt{201})}^{2}$ as $(3 + \sqrt{201}) (3 + \sqrt{201})$ .

$\frac{3 ((3 + \sqrt{201}) (3 + \sqrt{201}))}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Expand $(3 + \sqrt{201}) (3 + \sqrt{201})$ using the FOIL Method.

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Apply the distributive property.

$\frac{3 (3 (3 + \sqrt{201}) + \sqrt{201} (3 + \sqrt{201}))}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 \sqrt{201} + \sqrt{201} (3 + \sqrt{201}))}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Apply the distributive property.

$\frac{3 (3 \cdot 3 + 3 \sqrt{201} + \sqrt{201} \cdot 3 + \sqrt{201} \sqrt{201})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $3$ .

$\frac{3 (9 + 3 \sqrt{201} + \sqrt{201} \cdot 3 + \sqrt{201} \sqrt{201})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Move $3$ to the left of $\sqrt{201}$ .

$\frac{3 (9 + 3 \sqrt{201} + 3 \cdot \sqrt{201} + \sqrt{201} \sqrt{201})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Combine using the product rule for radicals.

$\frac{3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{201 \cdot 201})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Multiply $201$ by $201$ .

$\frac{3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{40401})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Rewrite $40401$ as $201^{2}$ .

$\frac{3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{201^{2}})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + 201)}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Add $9$ and $201$ .

$\frac{3 (210 + 3 \sqrt{201} + 3 \sqrt{201})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Add $3 \sqrt{201}$ and $3 \sqrt{201}$ .

$\frac{3 (210 + 6 \sqrt{201})}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Cancel the common factor of $210 + 6 \sqrt{201}$ and $16$ .

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Factor $2$ out of $3 (210 + 6 \sqrt{201})$ .

$\frac{2 (3 (105 + 3 \sqrt{201}))}{16} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Cancel the common factors.

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Factor $2$ out of $16$ .

$\frac{2 (3 (105 + 3 \sqrt{201}))}{2 (8)} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Cancel the common factor.

$\frac{2 (3 (105 + 3 \sqrt{201}))}{2 \cdot 8} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Rewrite the expression.

$\frac{3 (105 + 3 \sqrt{201})}{8} + 6 (- \frac{3 + \sqrt{201}}{8}) - 12$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{3 + \sqrt{201}}{8}$ into the numerator.

$\frac{3 (105 + 3 \sqrt{201})}{8} + 6 \frac{- (3 + \sqrt{201})}{8} - 12$

Factor $2$ out of $6$ .

$\frac{3 (105 + 3 \sqrt{201})}{8} + 2 (3) \frac{- (3 + \sqrt{201})}{8} - 12$

Factor $2$ out of $8$ .

$\frac{3 (105 + 3 \sqrt{201})}{8} + 2 \cdot 3 \frac{- (3 + \sqrt{201})}{2 \cdot 4} - 12$

Cancel the common factor.

$\frac{3 (105 + 3 \sqrt{201})}{8} + 2 \cdot 3 \frac{- (3 + \sqrt{201})}{2 \cdot 4} - 12$

Rewrite the expression.

$\frac{3 (105 + 3 \sqrt{201})}{8} + 3 \frac{- (3 + \sqrt{201})}{4} - 12$

Combine $3$ and $\frac{- (3 + \sqrt{201})}{4}$ .

$\frac{3 (105 + 3 \sqrt{201})}{8} + \frac{3 (- (3 + \sqrt{201}))}{4} - 12$

Multiply $- 1$ by $3$ .

$\frac{3 (105 + 3 \sqrt{201})}{8} + \frac{- 3 (3 + \sqrt{201})}{4} - 12$

Move the negative in front of the fraction.

$\frac{3 (105 + 3 \sqrt{201})}{8} - \frac{3 (3 + \sqrt{201})}{4} - 12$

To write $- \frac{3 (3 + \sqrt{201})}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 (105 + 3 \sqrt{201})}{8} - \frac{3 (3 + \sqrt{201})}{4} \cdot \frac{2}{2} - 12$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 (3 + \sqrt{201})}{4}$ and $\frac{2}{2}$ .

$\frac{3 (105 + 3 \sqrt{201})}{8} - \frac{3 (3 + \sqrt{201}) \cdot 2}{4 \cdot 2} - 12$

Multiply $4$ by $2$ .

$\frac{3 (105 + 3 \sqrt{201})}{8} - \frac{3 (3 + \sqrt{201}) \cdot 2}{8} - 12$

Combine the numerators over the common denominator.

$\frac{3 (105 + 3 \sqrt{201}) - 3 (3 + \sqrt{201}) \cdot 2}{8} - 12$

Simplify the numerator.

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Apply the distributive property.

$\frac{3 \cdot 105 + 3 (3 \sqrt{201}) - 3 (3 + \sqrt{201}) \cdot 2}{8} - 12$

Multiply $3$ by $105$ .

$\frac{315 + 3 (3 \sqrt{201}) - 3 (3 + \sqrt{201}) \cdot 2}{8} - 12$

Multiply $3$ by $3$ .

$\frac{315 + 9 \sqrt{201} - 3 (3 + \sqrt{201}) \cdot 2}{8} - 12$

Apply the distributive property.

$\frac{315 + 9 \sqrt{201} + (- 3 \cdot 3 - 3 \sqrt{201}) \cdot 2}{8} - 12$

Multiply $- 3$ by $3$ .

$\frac{315 + 9 \sqrt{201} + (- 9 - 3 \sqrt{201}) \cdot 2}{8} - 12$

Apply the distributive property.

$\frac{315 + 9 \sqrt{201} - 9 \cdot 2 - 3 \sqrt{201} \cdot 2}{8} - 12$

Multiply $- 9$ by $2$ .

$\frac{315 + 9 \sqrt{201} - 18 - 3 \sqrt{201} \cdot 2}{8} - 12$

Multiply $2$ by $- 3$ .

$\frac{315 + 9 \sqrt{201} - 18 - 6 \sqrt{201}}{8} - 12$

Subtract $18$ from $315$ .

$\frac{297 + 9 \sqrt{201} - 6 \sqrt{201}}{8} - 12$

Subtract $6 \sqrt{201}$ from $9 \sqrt{201}$ .

$\frac{297 + 3 \sqrt{201}}{8} - 12$

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{297 + 3 \sqrt{201}}{8} - 12 \cdot \frac{8}{8}$

Combine fractions.

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Combine $- 12$ and $\frac{8}{8}$ .

$\frac{297 + 3 \sqrt{201}}{8} + \frac{- 12 \cdot 8}{8}$

Combine the numerators over the common denominator.

$\frac{297 + 3 \sqrt{201} - 12 \cdot 8}{8}$

Simplify the numerator.

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Multiply $- 12$ by $8$ .

$\frac{297 + 3 \sqrt{201} - 96}{8}$

Subtract $96$ from $297$ .

$\frac{201 + 3 \sqrt{201}}{8}$

$x = - \frac{3 + \sqrt{201}}{8}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{3 + \sqrt{201}}{8}$ is a local minimum

Find the y-value when $x = - \frac{3 + \sqrt{201}}{8}$ .

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Replace the variable $x$ with $- \frac{3 + \sqrt{201}}{8}$ in the expression.

$f (- \frac{3 + \sqrt{201}}{8}) = {(- \frac{3 + \sqrt{201}}{8})}^{4} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Simplify the result.

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 + \sqrt{201}}{8}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = {(- 1)}^{4} {(\frac{3 + \sqrt{201}}{8})}^{4} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Apply the product rule to $\frac{3 + \sqrt{201}}{8}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = {(- 1)}^{4} (\frac{{(3 + \sqrt{201})}^{4}}{8^{4}}) + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $4$ .

$f (- \frac{3 + \sqrt{201}}{8}) = 1 (\frac{{(3 + \sqrt{201})}^{4}}{8^{4}}) + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $\frac{{(3 + \sqrt{201})}^{4}}{8^{4}}$ by $1$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{{(3 + \sqrt{201})}^{4}}{8^{4}} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $8$ to the power of $4$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{{(3 + \sqrt{201})}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Use the Binomial Theorem.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{3^{4} + 4 \cdot (3^{3} \sqrt{201}) + 6 \cdot (3^{2} {\sqrt{201}}^{2}) + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Simplify each term.

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Raise $3$ to the power of $4$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 4 \cdot (3^{3} \sqrt{201}) + 6 \cdot (3^{2} {\sqrt{201}}^{2}) + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $3$ to the power of $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 4 \cdot (27 \sqrt{201}) + 6 \cdot (3^{2} {\sqrt{201}}^{2}) + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $4$ by $27$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 6 \cdot (3^{2} {\sqrt{201}}^{2}) + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $3$ to the power of $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 6 \cdot (9 {\sqrt{201}}^{2}) + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $6$ by $9$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 54 {\sqrt{201}}^{2} + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 54 {(201^{\frac{1}{2}})}^{2} + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 54 \cdot 201^{\frac{1}{2} \cdot 2} + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 54 \cdot 201^{\frac{2}{2}} + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

Divide $1$ by $1$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 54 \cdot 201 + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Evaluate the exponent.

Multiply $54$ by $201$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 4 \cdot (3 {\sqrt{201}}^{3}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $4$ by $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 12 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{3}$ as ${(201^{3})}^{\frac{1}{2}}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{201^{3}} + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $201$ to the power of $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{8120601} + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Factor $40401$ out of $8120601$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{40401 (201)} + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite $40401$ as $201^{2}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{201^{2} \cdot 201} + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Pull terms out from under the radical.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 12 (201 \sqrt{201}) + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $201$ by $12$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + {\sqrt{201}}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{4}$ as $201^{2}$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + {(201^{\frac{1}{2}})}^{4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{1}{2} \cdot 4}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Combine $\frac{1}{2}$ and $4$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{4}{2}}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor of $4$ and $2$ .

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Factor $2$ out of $4$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2}}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factors.

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Factor $2$ out of $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 (1)}}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite the expression.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2}{1}}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Divide $2$ by $1$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{2}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $201$ to the power of $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 40401}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Add $81$ and $10854$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{10935 + 108 \sqrt{201} + 2412 \sqrt{201} + 40401}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Add $10935$ and $40401$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{51336 + 108 \sqrt{201} + 2412 \sqrt{201}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Add $108 \sqrt{201}$ and $2412 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{51336 + 2520 \sqrt{201}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor of $51336 + 2520 \sqrt{201}$ and $4096$ .

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Factor $8$ out of $51336$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{8 (6417) + 2520 \sqrt{201}}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Factor $8$ out of $2520 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{8 (6417) + 8 (315 \sqrt{201})}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Factor $8$ out of $8 (6417) + 8 (315 \sqrt{201})$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{8 (6417 + 315 \sqrt{201})}{4096} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factors.

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Factor $8$ out of $4096$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{8 (6417 + 315 \sqrt{201})}{8 \cdot 512} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{8 (6417 + 315 \sqrt{201})}{8 \cdot 512} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite the expression.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} + {(- \frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 + \sqrt{201}}{8}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} + {(- 1)}^{3} {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Apply the product rule to $\frac{3 + \sqrt{201}}{8}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} + {(- 1)}^{3} (\frac{{(3 + \sqrt{201})}^{3}}{8^{3}}) - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $- 1$ to the power of $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{{(3 + \sqrt{201})}^{3}}{8^{3}} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $8$ to the power of $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{{(3 + \sqrt{201})}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Use the Binomial Theorem.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{3^{3} + 3 \cdot (3^{2} \sqrt{201}) + 3 \cdot (3 {\sqrt{201}}^{2}) + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Simplify each term.

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Raise $3$ to the power of $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 3 \cdot (3^{2} \sqrt{201}) + 3 \cdot (3 {\sqrt{201}}^{2}) + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $3$ by $3^{2}$ by adding the exponents.

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Multiply $3$ by $3^{2}$ .

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Raise $3$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 3^{1 + 2} \sqrt{201} + 3 \cdot (3 {\sqrt{201}}^{2}) + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Add $1$ and $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 3^{3} \sqrt{201} + 3 \cdot (3 {\sqrt{201}}^{2}) + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $3$ to the power of $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 3 \cdot (3 {\sqrt{201}}^{2}) + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $3$ by $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 {(201^{\frac{1}{2}})}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201^{\frac{1}{2} \cdot 2} + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Divide $1$ by $1$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201 + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Evaluate the exponent.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201 + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Multiply $9$ by $201$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + {\sqrt{201}}^{3}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite ${\sqrt{201}}^{3}$ as ${(201^{3})}^{\frac{1}{2}}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{201^{3}}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Raise $201$ to the power of $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{8120601}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Factor $40401$ out of $8120601$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{40401 (201)}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite $40401$ as $201^{2}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{201^{2} \cdot 201}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Pull terms out from under the radical.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + 201 \sqrt{201}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Add $27$ and $1809$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{1836 + 27 \sqrt{201} + 201 \sqrt{201}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Add $27 \sqrt{201}$ and $201 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{1836 + 228 \sqrt{201}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor of $1836 + 228 \sqrt{201}$ and $512$ .

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Factor $4$ out of $1836$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459) + 228 \sqrt{201}}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Factor $4$ out of $228 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459) + 4 (57 \sqrt{201})}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Factor $4$ out of $4 (459) + 4 (57 \sqrt{201})$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459 + 57 \sqrt{201})}{512} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factors.

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Factor $4$ out of $512$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459 + 57 \sqrt{201})}{4 \cdot 128} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Cancel the common factor.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459 + 57 \sqrt{201})}{4 \cdot 128} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Rewrite the expression.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 {(- \frac{3 + \sqrt{201}}{8})}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{3 + \sqrt{201}}{8}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 ({(- 1)}^{2} {(\frac{3 + \sqrt{201}}{8})}^{2})$

Apply the product rule to $\frac{3 + \sqrt{201}}{8}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 ({(- 1)}^{2} (\frac{{(3 + \sqrt{201})}^{2}}{8^{2}}))$

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 (1 (\frac{{(3 + \sqrt{201})}^{2}}{8^{2}}))$

Multiply $\frac{{(3 + \sqrt{201})}^{2}}{8^{2}}$ by $1$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 \frac{{(3 + \sqrt{201})}^{2}}{8^{2}}$

Raise $8$ to the power of $2$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 \frac{{(3 + \sqrt{201})}^{2}}{64}$

Cancel the common factor of $2$ .

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Factor $2$ out of $- 6$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + 2 (- 3) (\frac{{(3 + \sqrt{201})}^{2}}{64})$

Factor $2$ out of $64$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + 2 \cdot (- 3 \frac{{(3 + \sqrt{201})}^{2}}{2 \cdot 32})$

Cancel the common factor.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + 2 \cdot (- 3 \frac{{(3 + \sqrt{201})}^{2}}{2 \cdot 32})$

Rewrite the expression.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 3 \frac{{(3 + \sqrt{201})}^{2}}{32}$

Combine $- 3$ and $\frac{{(3 + \sqrt{201})}^{2}}{32}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 {(3 + \sqrt{201})}^{2}}{32}$

Rewrite ${(3 + \sqrt{201})}^{2}$ as $(3 + \sqrt{201}) (3 + \sqrt{201})$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 ((3 + \sqrt{201}) (3 + \sqrt{201}))}{32}$

Expand $(3 + \sqrt{201}) (3 + \sqrt{201})$ using the FOIL Method.

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Apply the distributive property.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (3 (3 + \sqrt{201}) + \sqrt{201} (3 + \sqrt{201}))}{32}$

Apply the distributive property.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 \sqrt{201} + \sqrt{201} (3 + \sqrt{201}))}{32}$

Apply the distributive property.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 \sqrt{201} + \sqrt{201} \cdot 3 + \sqrt{201} \sqrt{201})}{32}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + \sqrt{201} \cdot 3 + \sqrt{201} \sqrt{201})}{32}$

Move $3$ to the left of $\sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \cdot \sqrt{201} + \sqrt{201} \sqrt{201})}{32}$

Combine using the product rule for radicals.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{201 \cdot 201})}{32}$

Multiply $201$ by $201$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{40401})}{32}$

Rewrite $40401$ as $201^{2}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{201^{2}})}{32}$

Pull terms out from under the radical, assuming positive real numbers.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + 201)}{32}$

Add $9$ and $201$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (210 + 3 \sqrt{201} + 3 \sqrt{201})}{32}$

Add $3 \sqrt{201}$ and $3 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (210 + 6 \sqrt{201})}{32}$

Cancel the common factor of $210 + 6 \sqrt{201}$ and $32$ .

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Factor $2$ out of $- 3 (210 + 6 \sqrt{201})$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 + 3 \sqrt{201}))}{32}$

Cancel the common factors.

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Factor $2$ out of $32$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 + 3 \sqrt{201}))}{2 (16)}$

Cancel the common factor.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 + 3 \sqrt{201}))}{2 \cdot 16}$

Rewrite the expression.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (105 + 3 \sqrt{201})}{16}$

Move the negative in front of the fraction.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - \frac{3 (105 + 3 \sqrt{201})}{16}$

To write $- \frac{459 + 57 \sqrt{201}}{128}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} \cdot \frac{4}{4} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Write each expression with a common denominator of $512$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{459 + 57 \sqrt{201}}{128}$ and $\frac{4}{4}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Multiply $128$ by $4$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Combine the numerators over the common denominator.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201} - (459 + 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Simplify the numerator.

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Apply the distributive property.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201} + (- 1 \cdot 459 - (57 \sqrt{201})) \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Multiply $- 1$ by $459$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201} + (- 459 - (57 \sqrt{201})) \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Multiply $57$ by $- 1$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201} + (- 459 - 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Apply the distributive property.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201} - 459 \cdot 4 - 57 \sqrt{201} \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Multiply $- 459$ by $4$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201} - 1836 - 57 \sqrt{201} \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Multiply $4$ by $- 57$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201}}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Subtract $1836$ from $6417$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 315 \sqrt{201} - 228 \sqrt{201}}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Subtract $228 \sqrt{201}$ from $315 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201}}{512} - \frac{3 (105 + 3 \sqrt{201})}{16}$

To write $- \frac{3 (105 + 3 \sqrt{201})}{16}$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201}}{512} - \frac{3 (105 + 3 \sqrt{201})}{16} \cdot \frac{32}{32}$

Write each expression with a common denominator of $512$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 (105 + 3 \sqrt{201})}{16}$ and $\frac{32}{32}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201}}{512} - \frac{3 (105 + 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Multiply $16$ by $32$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201}}{512} - \frac{3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Combine the numerators over the common denominator.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201} - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Simplify the numerator.

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Apply the distributive property.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201} + (- 3 \cdot 105 - 3 (3 \sqrt{201})) \cdot 32}{512}$

Multiply $- 3$ by $105$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201} + (- 315 - 3 (3 \sqrt{201})) \cdot 32}{512}$

Multiply $3$ by $- 3$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201} + (- 315 - 9 \sqrt{201}) \cdot 32}{512}$

Apply the distributive property.

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201} - 315 \cdot 32 - 9 \sqrt{201} \cdot 32}{512}$

Multiply $- 315$ by $32$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201} - 10080 - 9 \sqrt{201} \cdot 32}{512}$

Multiply $32$ by $- 9$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{4581 + 87 \sqrt{201} - 10080 - 288 \sqrt{201}}{512}$

Subtract $10080$ from $4581$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{- 5499 + 87 \sqrt{201} - 288 \sqrt{201}}{512}$

Subtract $288 \sqrt{201}$ from $87 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{- 5499 - 201 \sqrt{201}}{512}$

Simplify with factoring out.

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Rewrite $- 5499$ as $- 1 (5499)$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{- 1 \cdot 5499 - 201 \sqrt{201}}{512}$

Factor $- 1$ out of $- 201 \sqrt{201}$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{- 1 \cdot 5499 - (201 \sqrt{201})}{512}$

Factor $- 1$ out of $- 1 (5499) - (201 \sqrt{201})$ .

$f (- \frac{3 + \sqrt{201}}{8}) = \frac{- 1 (5499 + 201 \sqrt{201})}{512}$

Move the negative in front of the fraction.

$f (- \frac{3 + \sqrt{201}}{8}) = - \frac{5499 + 201 \sqrt{201}}{512}$

The final answer is $- \frac{5499 + 201 \sqrt{201}}{512}$ .

$y = - \frac{5499 + 201 \sqrt{201}}{512}$

These are the local extrema for $f (x) = x^{4} + x^{3} - 6 x^{2}$ .

$(0, 0)$ is a local maxima

$(- \frac{3 - \sqrt{201}}{8}, - \frac{5499 - 201 \sqrt{201}}{512})$ is a local minima

$(- \frac{3 + \sqrt{201}}{8}, - \frac{5499 + 201 \sqrt{201}}{512})$ is a local minima

Find the Local Maxima and Minima f(x)=x^4+x^3-6x^2

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