# Find the Local Maxima and Minima f(x)=x^4-8x^2+1

Find the first derivative of the function.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
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 .
Find the second derivative of the function.
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 .
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.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Factor.
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Remove unnecessary parentheses.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Set equal to and solve for .
Set the factor equal to .
Subtract from both sides of the equation.
Set equal to and solve for .
Set the factor equal to .
Add to both sides of the equation.
The solution is the result of and .
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.
Simplify each term.
Raising to any positive power yields .
Multiply by .
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 .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raising to any positive power yields .
Raising to any positive power yields .
Multiply by .
Simplify by adding zeros.
Add and .
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.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
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 .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Raise to the power of .
Multiply by .
Simplify by adding and subtracting.
Subtract from .
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.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
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 .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Raise to the power of .
Multiply by .
Simplify by adding and subtracting.
Subtract from .
Add and .
The final answer is .
These are the local extrema for .
is a local maxima
is a local minima
is a local minima
Find the Local Maxima and Minima f(x)=x^4-8x^2+1

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