Global Maximum and Minimum on $\mathbb{R}^n$ ¶

Definition¶

Let $A\subseteq\mathbb{R}^n$ $R^{n}$ , let $f\colon A\to\mathbb{R}$ $R$ , and let $\boldsymbol{x}^\ast\in A$ $A$ .

$f(\boldsymbol{x}^\ast)$ is a global maximum if $f(\boldsymbol{x}^\ast)\ge f(\boldsymbol{x})$ for all $\boldsymbol{x}\in A$ $A$ .
$f(\boldsymbol{x}^\ast)$ is a global minimum if $f(\boldsymbol{x}^\ast)\le f(\boldsymbol{x})$ for all $\boldsymbol{x}\in A$ $A$ .

The extremum is strict if the inequality is strict for all $\boldsymbol{x}\in A\setminus\{\boldsymbol{x}^\ast\}$ ${x^{*}}$ .

Interpretation¶

A global extremum compares the candidate with every feasible point. Strictness means there is no other feasible point with the same optimal value.

Graph of a function on the interval from 0 to 6 with a local maximum at a, a local minimum at b, the global maximum at c and the global minimum at the boundary point 0.

In the figure, the global minimum occurs at the boundary point $0$ and the global maximum at $c$ . The points $a$ and $b$ are only a local maximum and minimum, respectively, because their comparisons use a neighbourhood rather than the whole feasible set.

The extreme value theorem on compact sets gives an existence condition: a continuous function on a nonempty compact feasible set attains both extrema. The definition itself does not guarantee attainment on an arbitrary set.

Key Results¶

A strict global maximum or minimum has a unique optimizer.
A continuous function on a compact feasible set attains both a global maximum and a global minimum.
A global extremum is automatically local when neighbourhoods inside the feasible set are considered.

Examples¶

For $f(x,y)=-(x^2+y^2)$ on $\mathbb{R}^2$ , $(0,0)$ is a strict global maximizer.
For $f(x,y)=x^2$ on $\mathbb{R}^2$ , every point with $x=0$ is a global minimizer.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 4, Lecture 1

Keywords: global maximum, global minimum, global maximizer, global minimizer, absolute extremum Symbols: element of, real numbers, subset

Last updated: 17 October 2026 21:53

Global Maximum and Minimum on Rn\mathbb{R}^nRn¶