Local 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 local maximum if there is a neighbourhood $U$ of $\boldsymbol{x}^\ast$ such that $f(\boldsymbol{x}^\ast)\ge f(\boldsymbol{x})$ for all $\boldsymbol{x}\in U\cap A$ $A$ .
$f(\boldsymbol{x}^\ast)$ is a local minimum if there is a neighbourhood $U$ of $\boldsymbol{x}^\ast$ such that $f(\boldsymbol{x}^\ast)\le f(\boldsymbol{x})$ for all $\boldsymbol{x}\in U\cap A$ $A$ .

The extremum is strict if the inequality is strict for all nearby feasible points other than $\boldsymbol{x}^\ast$ .

Interpretation¶

A local extremum is best only against nearby feasible alternatives. It may or may not be globally best.

Two families of circular level sets: values decrease outwards around a local maximum and increase outwards around a local minimum; a dashed neighbourhood U is drawn around each point.

The red level sets decrease in value as one moves away from the local maximum, while the blue ones increase away from the local minimum. Only nearby feasible points in $U\cap A$ are compared. An interior local extremum of a $C^1$ function is stationary, but the Hessian may be inconclusive when it is semidefinite.

The marked critical points are classified by their Hessians: with the colours of the legend for minima, maxima and saddles.

Marked points are critical points, classified by the Hessian; move $\boldsymbol{x}_0$ onto one and the tangent plane becomes horizontal.

An open metric ball can specify the neighbourhood: choose a radius $r>0$ and compare the candidate only with feasible points less than $r$ away. Intersecting the ball with the feasible set also covers candidates on its boundary.

Key Results¶

Interior local extrema of differentiable unconstrained problems must satisfy a first-order necessary condition.
Second-order conditions use the Hessian to classify critical points when possible.
Local extrema need not be unique and need not be global.

Examples¶

For $f(x,y)=x^2+y^2$ $y^{2}$ , $(0,0)$ is a strict local minimum.
For $f(x)=x^3-3x$ $3 x$ , $x=-1$ is a local maximum and $x=1$ is a local minimum.

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 4, Lecture 1

Keywords: local maximum, local minimum, local maximizer, local minimizer, relative extremum Symbols: element of, intersection, real numbers, subset

Last updated: 17 October 2026 21:53

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