# Calculus Examples

,

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 .

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

Add and .

Set the first derivative equal to zero.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Divide by .

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 .

Use the endpoints and all critical points on the interval to test for any absolute extrema over the given interval.

Simplify each term.

Remove parentheses.

Raising to any positive power yields .

Remove parentheses.

Raising to any positive power yields .

Multiply by .

Simplify by adding zeros.

Add and .

Subtract from .

Simplify each term.

Remove parentheses.

Raise to the power of .

Remove parentheses.

Raise to the power of .

Multiply by .

Simplify by subtracting numbers.

Subtract from .

Subtract from .

Simplify each term.

Raise to the power of .

Multiply by by adding the exponents.

Combine and

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Raise to the power of .

Simplify by subtracting numbers.

Subtract from .

Subtract from .

Simplify each term.

Remove parentheses.

Raise to the power of .

Remove parentheses.

Raise to the power of .

Multiply by .

Simplify by subtracting numbers.

Subtract from .

Subtract from .

Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.

Absolute Maximum:

Absolute Minimum: