Extreme value

An extreme value, or extremum (plural extrema), is the smallest (minimum) or largest (maximum) value of a function, either in an arbitrarily small neighborhood of a point in the function's domain — in which case it is called a relative or local extremum — or on a given set contained in the domain (perhaps all of it) — in which case it is called an absolute or global extremum (the latter term is common when the set is all of the domain).

As a special case, an extremum that would otherwise be considered a relative/local extremum but occurs at an endpoint (or more generally a boundary) of the function's domain is sometimes called an endpoint or boundary extremum and is not considered a relative/local extremum, although it may be an absolute/global one.

Note that in the case of relative/local extrema, it is common to concentrate on where the extrema occur (i.e., the "x-values") rather than what the extreme values actually are (the "y-values"), whereas in the case of absolute/global extrema it is common to concentrate on the extreme value itself (the "y-value"). However, in either case both values may be given — e.g., $f(2)=5$ if the extreme value 5 occurs at $x = 2$ .

Extrema can be found by taking the derivative of a function and setting it to equal zero.

Definitions

For a real-valued function of a single real variable

Given $f:\Re \to \Re$ ,

$f$ achieves a relative maximum (or local maximum) at $x_0$ if there is some open interval $I$ containing $x_0$ for which $f(x)\le f(x_0)$ for all $x$ on $I$ ;
$f$ achieves a relative minimum (or local minimum) at $x_0$ if there is some open interval $I$ containing $x_0$ for which $f(x)\ge f(x_0)$ for all $x$ on $I$ ;
$f$ achieves its absolute maximum (or global maximum) value $f(x_0)$ on a set $D$ if $\vec x_0\in D$ and $f(x)\le f(x_0)$ for all $x$ in $D$ ;
$f$ achieves its absolute minimum (or global minimum) value $f(x_0)$ on a set $D$ if $\vec x_0\in D$ and $f(x)\ge f(x_0)$ for all $x$ in $D$ .

Note also that a relative/local extremum cannot happen at an endpoint of the function's domain.

For a real-valued function of more than one real variable

Given $f:\Re ^{n}\to \Re$ , for some integer $n>1$ ,

$f$ achieves a relative maximum (or local maximum) at $\vec x_0$ if there is some open ball $B$ containing $\vec x_0$ for which $f(\vec x)\le f(\vec x_0)$ for all $\vec{x}$ in $B$ ;
$f$ achieves a relative minimum (or local minimum) at $\vec x_0$ if there is some open ball $B$ containing $\vec x_0$ for which $f(\vec x)\ge f(\vec x_0)$ for all $\vec{x}$ in $B$ ;
$f$ achieves its absolute maximum (or global maximum) value $f(\vec x_0)$ on a set $D$ if $\vec x_0\in D$ and $f(\vec x)\le f(\vec x_0)$ for all $\vec{x}$ in $D$ ;
$f$ achieves its absolute minimum (or global minimum) value $f(\vec x_0)$ on a set $D$ if $\vec x_0\in D$ and $f(\vec x)\ge f(\vec x_0)$ for all $\vec{x}$ in $D$ .