Point of absolute extremum

Definition

A point of absolute extremum refers to a point in the domain of a function that is either a point of absolute maximum or a point of absolute minimum. Both these are defined below.

Point of absolute maximum

A point $c$ in the domain of a function $f$ is defined as a point of absolute maximum if $f(x)\le f(c)$ for all $x$ in the domain of the function.

The value $f(c)$ is termed the absolute maximum value.

Note that for a given function, it is possible for there to be more than one point of absolute maximum, but the absolute maximum value must be the same at all points of absolute maximum.

Point of absolute minimum

A point $c$ in the domain of a function $f$ is defined as a point of absolute minimum if $f(x)\ge f(c)$ for all $x$ in the domain of the function.

The value $f(c)$ is termed the absolute minimum value.

Note that for a given function, it is possible for there to be more than one point of absolute minimum, but the absolute minimum value must be the same at all points of absolute minimum.