Write as a function.

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 .

Move to the left of .

Rewrite as .

Rewrite as .

Multiply by .

Subtract from .

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

Differentiate using the Product Rule which states that is where and .

Differentiate.

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

Differentiate using the Power Rule which states that is where .

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 .

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 .

Differentiate using the Power Rule which states that is where .

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

Simplify the expression.

Add and .

Multiply by .

Simplify.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Combine terms.

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Multiply by .

Move to the left of .

Multiply by .

Multiply by .

Multiply by .

Add and .

Add and .

Subtract from .

Multiply by .

Multiply by .

Subtract from .

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 .

Rewrite as .

Factor out of .

Factor out of .

Factor.

Factor by grouping.

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Multiply by .

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

Remove unnecessary parentheses.

Multiply each term in by .

Simplify .

Simplify by multiplying through.

Apply the distributive property.

Multiply.

Multiply by .

Multiply by .

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 by adding the exponents.

Move .

Multiply by .

Multiply by .

Rewrite as .

Multiply by .

Subtract from .

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Multiply by .

Multiply by .

For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .

Factor out of .

Rewrite as plus

Apply the distributive property.

Multiply by .

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

Factor out the greatest common factor (GCF) from each group.

Factor the polynomial by factoring out the greatest common factor, .

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set the first factor equal to .

Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Move the negative in front of the fraction.

Set the next factor equal to .

Add to both sides of the equation.

The final solution is all the values that make true.

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 .

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.

Write as a fraction with a common denominator.

Combine the numerators over the common denominator.

Add and .

To write as a fraction with a common denominator, multiply by .

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

Move the negative in front of the fraction.

Use the power rule to distribute the exponent.

Apply the product rule to .

Apply the product rule to .

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Raise to the power of .

Raise to the power of .

Raise to the power of .

Multiply .

Multiply and .

Multiply by .

Multiply by .

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.

Multiply by .

Add and .

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.

Add and .

Multiply by .

Subtract from .

Raising to any positive power yields .

Multiply by .

The final answer is .

These are the local extrema for .

is a local minima

is a local maxima

Find the Local Maxima and Minima -(x+1)(x-1)^2