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# Local extrema

Given a function $f$, a point $\xo$ is

• a local maximum if, for all $x$ around $\xo$, $$f(x)\le f(\xo);$$
• a local minimum if, for all $x$ around $\xo$, $$f(x)\ge f(\xo).$$

An extremum is either a minimum or a maximum. A local extremum is sometimes called a turning point or a relative extremum.

$0$ is a minimum of $x^2$ and a maximum of $-x^2$.

At a local extremum, the tangent must be horizontal $$f'(\xo)=0.$$

At a local maximum $\xo$, $f$ "increases" to its left, so $f'(\xo)\ge 0$, and "decreases" to its right, so $f'(\xo)\le 0$. Thus, $f'(\xo)=0$.

Local minimum (A), inflection point (B), local maximum (C) of a function