# Extreme value

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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. If the second derivative at this point is positive, it is a minimum, and vice versa.

## DefinitionsEdit

### For a real-valued function of a single real variableEdit

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 variableEdit

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 $D$.

Here $\vec x$ is a vector representing the n-tuple $(x_1,x_2,\ldots,x_n)\in\operatorname{dom}f$.

Note that a relative/local extremum cannot happen on the boundary of the function's domain.