By the Sum Rule, the derivative of with respect to is .

Evaluate .

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 .

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 .

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.

Since is constant with respect to , the derivative of with respect to is .

Add and .

By the Sum Rule, the derivative of with respect to is .

Evaluate .

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 .

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.

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 .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Divide each term in by .

Cancel the common factor of .

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 the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Rewrite as .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

Multiply by .

Simplify .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Rewrite as .

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 numerator.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Rewrite as .

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.

Simplify each term.

Cancel the common factor of .

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.

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

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Use the power rule to distribute the exponent.

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.

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 .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

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 .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

Multiply by .

Add and .

Subtract from .

Multiply .

Multiply by .

Combine and .

Use the power rule to distribute the exponent.

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.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Multiply .

Multiply by .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

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 .

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 .

Multiply and .

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

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 .

Multiply and .

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

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.

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

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.

Simplify each term.

Cancel the common factor of .

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.

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

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Use the power rule to distribute the exponent.

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.

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 .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

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 .

Factor out of .

Rewrite as .

Pull terms out from under the radical.

Multiply by .

Add and .

Add and .

Multiply .

Multiply by .

Combine and .

Use the power rule to distribute the exponent.

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.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Multiply .

Multiply by .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Use to rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

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 .

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 .

Multiply and .

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

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 .

Multiply and .

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

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.

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

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