Absolute Extremum

analysis

Definition

Absolute Extremum

An absolute extremum (or global extremum) of a function $f:D\to \mathbb{R}$ is either an absolute maximum or an absolute minimum.

• $x_0\in D$ is an absolute maximum if $f(x)\le f(x_0)$ for all $x\in D$.
• $x_0\in D$ is an absolute minimum if $f(x)\ge f(x_0)$ for all $x\in D$.

Existence

If $f$ is a continuous function on a compact set (such as a closed and bounded interval $[a,b]$), then the existence of both an absolute maximum and an absolute minimum is guaranteed.