Local extremum

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 $x$.

Local extrema are closely connected to critical points and the derivative (when it exists). Criteria such as the second derivative tests help distinguish maxima from minima.

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.