AP Calculus AB and BC glossary
Extreme Value Theorem
Also called: EVT
The Extreme Value Theorem says that a function continuous on a closed, bounded interval must attain both an absolute maximum and an absolute minimum somewhere on that interval. It guarantees the extrema exist without saying where they are.
Both hypotheses carry weight. On the open interval the function attains neither extreme, and a discontinuous function on a closed interval can also miss them. Drop either condition and the guarantee is gone.
Because the EVT tells you the extrema exist and must sit at a critical point or an endpoint, it is what licenses the candidates test: evaluate the function at every critical point and both endpoints, then compare.
Appears in: Unit 5: Analytical Applications