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

Math
Write as a function.
Find the first derivative of the function.
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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 .
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.
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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.
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Add and .
Multiply by .
Simplify.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Combine terms.
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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 .
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 the left side of the equation.
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Factor out of .
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Factor out of .
Factor out of .
Rewrite as .
Factor out of .
Factor out of .
Factor.
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Factor by grouping.
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For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by .
Factor out the greatest common factor from each group.
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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
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Multiply each term in by .
Simplify .
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Simplify by multiplying through.
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Apply the distributive property.
Multiply.
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Multiply by .
Multiply by .
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 by adding the exponents.
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Move .
Multiply by .
Multiply by .
Rewrite as .
Multiply by .
Subtract from .
Apply the distributive property.
Simplify.
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Multiply by .
Multiply by .
Multiply by .
Multiply by .
Factor by grouping.
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For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Factor out of .
Rewrite as plus
Apply the distributive property.
Multiply by .
Factor out the greatest common factor from each group.
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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 and solve.
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Set the first factor equal to .
Subtract from both sides of the equation.
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 .
Move the negative in front of the fraction.
Set the next factor equal to and solve.
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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.
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 .
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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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.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Use the power rule to distribute the exponent.
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Apply the product rule to .
Apply the product rule to .
Multiply by by adding the exponents.
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Move .
Multiply by .
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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 .
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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.
Evaluate the second derivative.
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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
Find the y-value when .
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Replace the variable with in the expression.
Simplify the result.
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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

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