Local and Absolute Maxima and Minima — Cheat sheet

CHEAT SHEET · FREE

Key method

An absolute minimum at $a$ satisfies $f(a)\le f(x)$ for every allowed $x$ . A local minimum needs this only in a neighborhood of $a$ within the domain. Maximum reverses the inequality.

Example

For $f(x)=(x-1)^2$ on $[0,3]$ , the absolute minimum is $0$ at $x=1$ . Endpoint values are $1$ and $4$ , so the absolute maximum is $4$ at $x=3$ . On the open interval $(0,3)$ , values approach $4$ but never attain it, so no absolute maximum exists.

Avoid this mistake

A bound need not be attained. The words “maximum” and “minimum” require actual inputs that produce those values.