Write as a 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 .

Combine and .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide 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 .

Combine and .

Multiply by .

Combine and .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide 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 .

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 .

To find the local maximum and minimum values of the function, set the derivative equal to and solve.

Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .

Write the factored form using these integers.

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 .

Add to both sides of the equation.

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.

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

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Raise to the power of .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Raise to the power of .

Multiply .

Multiply by .

Combine and .

Multiply by .

Move the negative in front of the fraction.

Multiply by .

Find the common denominator.

Write as a fraction with denominator .

Multiply by .

Multiply and .

Write as a fraction with denominator .

Multiply by .

Multiply and .

Write as a fraction with denominator .

Multiply by .

Multiply and .

Combine fractions.

Combine fractions with similar denominators.

Multiply.

Multiply by .

Multiply by .

Multiply by .

Simplify the numerator.

Subtract from .

Add and .

Subtract from .

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 .

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.

Raise to the power of .

Combine and .

Raise to the power of .

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 .

Multiply by .

Find the common denominator.

Write as a fraction with denominator .

Multiply by .

Multiply and .

Write as a fraction with denominator .

Multiply by .

Multiply and .

Write as a fraction with denominator .

Multiply by .

Multiply and .

Combine fractions.

Combine fractions with similar denominators.

Multiply.

Multiply by .

Multiply by .

Multiply by .

Simplify the numerator.

Subtract from .

Add and .

Subtract from .

The final answer is .

These are the local extrema for .

is a local minima

is a local maxima

Find the Local Maxima and Minima y=1/3x^3-5/2x^2+6x-3