Local extremum - Knowlpedia

A local extremum of a function $f:I\to\mathbb{R}$ at a point $a\in I$ means either a local maximum or a local minimum: $a$ is a local maximum if there exists $\delta>0$ such that $f(a)\ge f(x)$ for all $x\in I$ with $|x-a|<\delta$ , and a local minimum if there exists $\delta>0$ such that $f(a)\le f(x)$ for all such $x$ .

Examples

For $f(x)=x^2$ , the point $a=0$ is a local minimum.
For $f(x)=-x^2$ , the point $a=0$ is a local maximum.