Extremum

The (absolute) maximum and (absolute) minimumpl. maxima, minima. of a function refer to the greatest and least values of the range, respectively.[cite:@enwiki:1281908276@boelkins2025active] Instead of the entire range, extremumExtremum refers generically to a maximum or minimum. can also be considered relative to a given interval, in which case they are called the relative maximum or relative minimum; or to a neighbourhood, where they will be called local extrema.[cite:@enwiki:1281908276@boelkins2025active]

Nomenclature

Maximum and minimum (pl. maxima, minima) refer, respectively, to the greatest and least values. Extremum (pl. extrema) refers to either. Useful adjectives include maximal, minimal and extreme.

Definition

For some function $f$ and a member of the domain $c$, $fc$ is

• an (absolute) maximum if $fc⩾fx$ for every $x$,

• a local maximum if $fc⩾fx$ for all $x$ near $c$,

• and a relative maximum if $fc⩾fx$ for all $x$ in some interval containing $c$.[cite:@boelkins2025active]

Of course, swapping out the $⩾$ relation for $⩽$ gives us the definitions for absolute, local, and relative minima.

Local extrema

The local extrema of differentiable functions can only appear at "turning points" (critical points) of curves; i.e., zeroes of the derivative, but not all critical points are extrema; critical points that are not extrema are called saddle points.