local extremum

A point where a function's value is largest or smallest compared only to its immediate neighbors.

$f$ has a local maximum (minimum) at $c$ if $f(c) \ge f(x)$ ( $f(c) \le f(x)$ ) for every $x$ near $c$ . Fermat's theorem says every local extremum at an interior point where $f$ is differentiable must have $f'(c)=0$ .

Examples

$f(x)=x^3-3x$ has a local maximum at $x=-1$ (value $2$ ) and a local minimum at $x=1$ (value $-2$ ).