Determining Absolute (Global) Extrema on a Closed Interval

What Absolute Extrema Are

An absolute maximum, also called a global maximum, is the largest function value attained on a specified domain. An absolute minimum, or global minimum, is the smallest function value attained on that domain.

For a function restricted to a closed interval , the key question is not simply where the derivative is zero. You must compare the function values at all critical points in the interval and both endpoints.

This procedure is sometimes called the closed-interval method. It is one of the most important applications of derivatives in AP Calculus.

The Extreme Value Theorem

The reason closed intervals are so important comes from the Extreme Value Theorem.

If a function is continuous on a closed interval , then the function is guaranteed to have both an absolute maximum and an absolute minimum somewhere on that interval.

This theorem tells you that the extrema exist. The derivative-based closed-interval method tells you how to find them.

The Closed-Interval Method

To determine the absolute maximum and minimum of a continuous function on a closed interval, use these steps:

1. Find the derivative .
2. Find all critical numbers in the interval by solving and finding where does not exist.
3. Keep only the critical numbers that lie inside the interval.
4. Evaluate the original function at every critical number and at both endpoints.
5. Compare all resulting function values.
6. The largest value is the absolute maximum; the smallest value is the absolute minimum.

Worked Examples

Why You Must Check the Endpoints

A common mistake is to find the critical points and stop. That does not guarantee the absolute extrema on a closed interval.

An absolute maximum or minimum can occur at an endpoint even when the derivative is not zero there. Endpoints are not interior points, so the usual critical-number condition does not apply to them.

Absolute vs. Local Extrema

It is important to distinguish between local and absolute extrema.

• A local maximum is a function value that is greater than nearby function values.
• A local minimum is a function value that is smaller than nearby function values.
• An absolute maximum is the greatest function value anywhere in the specified domain.
• An absolute minimum is the smallest function value anywhere in the specified domain.

A function can have several local extrema but only one absolute maximum value and one absolute minimum value, although the same absolute value can sometimes occur at multiple x-values.

A Fast AP Calculus Strategy

For a closed-interval absolute-extrema problem, you can use this compact checklist:

Think of the process as building a short list of candidates. Critical numbers inside the interval are candidates, and the two endpoints are candidates. Once every candidate has been evaluated using the original function, the largest and smallest values settle the problem.

Common Mistakes

Practice Problems