Relative Extrema on an Interval

$f(x)$ has a relative (local) maximum at $x_0$ if there is an open interval containing $x_0$ on which $f(x_0)$ is the largest value; similarly, $f(x)$ has a relative (local) minimum at $x_0$ if there is an open interval containing $x_0$ on which $f(x_0)$ is the smallest value.

A relative maximum need not be the largest value on the entire domain (and a relative minimum need not be the overall smallest) — there may be several local maxima or minima across the domain. A relative extremum is defined only by comparison with nearby values of $f$, not the whole domain.

Theorem 7.9 (Fermat). If $f(x)$ has a relative extremum at $x=c$, then $c$ is a critical number. So every local extremum is found among the solutions of $f'(x)=0$ together with the points where $f'(x)$ fails to exist. …