Global Extremum¶

Definition¶

A function $f$ defined on a domain $X\subseteq \mathbb{R}$ attains its global maximum (aka absolute maximum) at $x=x_{\mathrm{max}}\text{,}$ if $f(x_{\mathrm{max}}) \geq f(x)$ for all $x\in X\text{.}$ Similarly, the function attains its global minimum (aka absolute minimum) at $x=x_{\mathrm{min}}\text{,}$ if $f(x_{\mathrm{min}}) \leq f(x)$ for all $x \in X\text{.}$

Interpretation¶

A global extremum describes the largest or smallest value a function attains over its entire domain. Unlike local extrema, which compare $f(x)$ only with values at nearby inputs, global extrema account for all possible inputs.

A cubic on an interval from a to b with a local maximum that is not global, a global minimum inside the interval, and the global maximum at the endpoint b.

On $[a,b]$ the cubic has a local maximum inside the interval, but it is not global: the value at the endpoint $b$ is larger. The global minimum lies inside the interval, at the bottom of a valley, so it is also a local minimum. The global maximum, by contrast, sits at an endpoint, where the graph is still rising.

Move the interval $[a,b]$ : the global maximum and minimum are found among the endpoints and the stationary points inside, and a local extremum need not be global.

The definition separates the location from the value. The point $x_{\mathrm{max}}$ need not be unique: $\cos x$ on $\mathbb{R}$ attains its global maximum $1$ at every integer multiple of $2\pi$ . The value $f(x_{\mathrm{max}})$ is unique when it exists, since it is the maximum of the range $\{f(x)\colon x\in X\}$ $X}$ . It can fail to exist even for a bounded function: $f(x)=x$ on $(0,1)$ has supremum $1$ but no maximum. The domain is part of the question, too: $x^2$ has no global maximum on $\mathbb{R}$ , but on $[-1,2]$ it has one at $x=2$ $2$ , with value $4$ .

For a continuous function on a closed bounded interval, the extreme value theorem guarantees that both global extrema exist. Math IV uses the same definition for functions of several variables.

Key Results¶

Global extrema may occur at interior points or boundary points of the domain.
A global extremum may not exist on unbounded or open domains.

Examples¶

The function $f(x)=x^2$ has a global minimum at $x=0\text{.}$
The function $f(x)=x$ has no global extremum on $\mathbb{R}\text{.}$

Prerequisites¶

Local Extremum

Sources¶

Math I: Calculus · Lecture 5

Symbols: element of, pi, real numbers, subset

Last updated: 20 October 2026 03:02