Local maximum and minimum

Definition

Let $D \subseteq \mathbb{R}^n$ , let $f \colon D \to \mathbb{R}$ , and let $a \in D$ . The function $f$ has a local maximum at $a$ if there is $\varepsilon > 0$ such that $f(x) \le f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$ . The function $f$ has a local minimum at $a$ if there is $\varepsilon > 0$ such that $f(x) \ge f(a)$ for all $x \in D$ with $\|x - a\| < \varepsilon$ . A local extremum of $f$ is a local maximum or a local minimum of $f$ . The extremum is strict if the inequality is strict for all such $x \ne a$ .

Remarks

A global maximum of $f$ is a point $a$ with $f(x) \le f(a)$ for all $x \in D$ . Each global maximum is a local maximum. The function $f(x, y) = x^2 + y^2$ has a strict local minimum at $(0, 0)$ , which is also a global minimum. In two or more variables, a local minimum that is the only critical point need not be a global minimum. For $f(x, y) = x^2 + y^2 (1 - x)^3$ , the only critical point $(0, 0)$ is a local minimum, but $f(2, 3) = -5 < f(0, 0)$ . At an interior local extremum of a differentiable function, the gradient is zero.

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Tag	`04CPG9K1C4`
URI	`m2://04CPG9K1C4`
Kind	definition
Writing	W0 stub
MSC	26B99
Defines	Local maximum and minimum
Uses	Function of several variables

Local maximum and minimum

Definition

Remarks

Reading path

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