Local minimum

Local minimum
Tag	`0NS8BP96QX`
URI	`m2://0NS8BP96QX`
Aliases	`relative-minimum`
Kind	definition
Writing	W0 stub
MSC	26A24
Defines	Local minimum
Uses	Local maximum

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Definition

Let $D \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ , and let $a \in D$ . The function $f$ has a local minimum at $a$ if there is $\delta > 0$ with $f(x) \ge f(a)$ for all $x \in D$ with $|x - a| < \delta$ . It is a strict local minimum if $f(x) > f(a)$ for all such $x \ne a$ .

Remarks

So $f$ has a local minimum at $a$ exactly when $-f$ has a local maximum at $a$ . For example, $x^2$ has a strict local minimum at $0$ . Mathlib defines it as IsLocalMin.

Explanations

L1 human

A local minimum is the bottom of a valley of the graph. Nearby values are not lower.

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Local maximum definition
Local minimum definition

What depends on this

79 nodes use this node, directly or through other nodes.

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