AP Calculus AB and BC glossary

Absolute minimum

Also called: Global minimum

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

The absolute minimum is the lowest output on the interval, the twin of the absolute maximum. The same Extreme Value Theorem guarantees both exist together when ff is continuous on a closed interval.

The procedure copies the maximum: list the critical numbers and endpoints, evaluate ff at each, and this time keep the smallest result. A minimum can occur at a sharp corner where f′f' is undefined, so include those points too.

The mistake

Assuming the minimum must be where f′(c)=0f'(c) = 0. On a closed interval the smallest value can sit at an endpoint or at a corner where the derivative does not exist, both of which a search for f′=0f' = 0 alone skips.

Appears in: Unit 5: Analytical Applications