AP Calculus AB and BC glossary

Absolute maximum

Also called: Global maximum

The absolute maximum is the largest value a function attains on a given interval. On a closed interval a continuous function is guaranteed one, and you find it with the Candidates Test: evaluate the function at every critical number and at both endpoints, then take the largest output.

The absolute maximum is an output value, the height of the highest point, not the xx where it happens. The Extreme Value Theorem guarantees one exists whenever ff is continuous on a closed interval [a,b][a, b].

To find it, build a short list: every critical number inside the interval plus the two endpoints. Evaluate ff at each and compare the numbers. The largest wins, and it often sits at an endpoint, where f′f' never signals a turn.

The mistake

Reporting the location instead of the value. The absolute maximum is the output f(c)f(c), not the xx-value cc where it occurs, and it is easy to miss one that lives at an endpoint rather than a critical number.

Appears in: Unit 5: Analytical Applications