Find the Local Maxima and Minima f(x)=-2x^3-14x^2+2x+84

Math
Find the first derivative of the function.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
Find the second derivative of the function.
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By the Sum Rule, the derivative of with respect to is .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Evaluate .
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Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Divide by .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Simplify the expression to solve for the portion of the .
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Simplify the numerator.
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Raise to the power of .
Multiply by .
Multiply by .
Add and .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
The final answer is the combination of both solutions.
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Cancel the common factor of .
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Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
Subtract from .
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Find the y-value when .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Use the power rule to distribute the exponent.
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Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
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Raise to the power of .
Raise to the power of .
Multiply by .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Add and .
Subtract from .
Multiply .
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Multiply by .
Combine and .
Use the power rule to distribute the exponent.
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Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply .
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Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Add and .
Subtract from .
Combine and .
Move the negative in front of the fraction.
Multiply .
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Multiply by .
Combine and .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Add and .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Add and .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
The final answer is .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
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Simplify each term.
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Cancel the common factor of .
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Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
Add and .
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Find the y-value when .
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
Tap for more steps…
Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Use the Binomial Theorem.
Simplify each term.
Tap for more steps…
Raise to the power of .
Raise to the power of .
Multiply by .
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Multiply by .
Apply the product rule to .
Raise to the power of .
Rewrite as .
Raise to the power of .
Rewrite as .
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Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Add and .
Add and .
Multiply .
Tap for more steps…
Multiply by .
Combine and .
Use the power rule to distribute the exponent.
Tap for more steps…
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Raise to the power of .
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by .
Multiply by .
Multiply by .
Multiply .
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Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Add and .
Add and .
Combine and .
Move the negative in front of the fraction.
Multiply .
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Multiply by .
Combine and .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Subtract from .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Tap for more steps…
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Apply the distributive property.
Multiply by .
Multiply by .
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Subtract from .
To write as a fraction with a common denominator, multiply by .
Combine fractions.
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Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps…
Multiply by .
Add and .
The final answer is .
These are the local extrema for .
is a local maxima
is a local minima
Find the Local Maxima and Minima f(x)=-2x^3-14x^2+2x+84

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