Local Extremum¶

Definition¶

Let $f$ be a function defined on a domain $X\subseteq \mathbb{R}\text{.}$ A point $x=x_{\mathrm{max}}$ is a local maximum point (aka relative maximum point) of $f$ if there exists some $\varepsilon > 0$ such that $f(x_{\mathrm{max}}) \geq f(x)$ for all $x \in X$ with $|x-x_{\mathrm{max}}|<\varepsilon\text{.}$ Similarly, $x=x_{\mathrm{min}}$ is a local minimum point (aka relative minimum point) of $f$ if there exists some $\varepsilon > 0$ such that $f(x_{\mathrm{min}}) \leq f(x)$ for all $x \in X$ with $|x-x_{\mathrm{min}}|<\varepsilon\text{.}$

Interpretation¶

A local extremum is a point where a function attains a maximum or minimum relative to nearby values: the comparison is made only with the points of the domain within some distance $\varepsilon$ , not with the whole domain. It captures optimal behaviour in a neighbourhood rather than globally.

The cubic x cubed minus 3x with a local maximum at -1 and a local minimum at 1, each inside a shaded vertical strip showing the neighbourhood that is compared.

The figure shows $f(x)=x^3-3x$ $3 x$ , with a local maximum at $x_{\max}=-1$ $- 1$ , where $f(-1)=2$ $2$ , and a local minimum at $x_{\min}=1$ $1$ , where $f(1)=-2$ $- 2$ . Inside each shaded strip of half-width $\varepsilon$ , the marked value is the largest, respectively the smallest, value of $f$ . Neither is global: $f(x)>2$ for $x>2$ and $f(x)<-2$ for $x<-2$ $- 2$ .

Grow the neighbourhood radius $\varepsilon$ : the comparison $f(x_{\max})\ge f(x)$ only has to hold for $x$ within $\varepsilon$ of the point, not on the whole domain. A local extremum only needs one $\varepsilon>0$ that works.

The definition only asks for some $\varepsilon>0$ $0$ , and a smaller $\varepsilon$ can only make the comparison easier. The inequality is not strict, so a constant function has a local maximum and a local minimum at every point. Only points of the domain are compared, which is why an endpoint can be a local extremum: $f(x)=x$ on $[0,1]$ has a local minimum at $0$ , although its one-sided derivative there is $1$ .

Every global extremum is a local one, but not conversely. To find local extrema at interior points, one first lists the critical points with the first-order condition and then checks each of them by comparing nearby values. The definition returns in Math IV for functions of several variables as a local maximum or minimum.

Key Results¶

Under this definition a local extremum may occur at an interior point or at a boundary point of the domain. The first-order condition applies only to interior points where $f$ is differentiable; boundary points must be examined separately.
Local extrema need not be global.
Critical points are candidates for local extrema.

Examples¶

The function $f(x)=x^2$ has a local minimum at $x=0\text{.}$
The function $f(x)=x^3$ has no local extremum.

Prerequisites¶

Function

Sources¶

Math I: Calculus · Lecture 6

Aliases: relative extremum Keywords: local maximum, local minimum, turning point Symbols: element of, epsilon, real numbers, subset

Last updated: 20 October 2026 03:02