# 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., if the extreme value 5 occurs at .

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..

## Contents

[show]## Definitions

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

Given ,

- achieves a
*relative maximum*(or*local maximum*) at if there is some open interval containing for which for all on ; - achieves a
*relative minimum*(or*local minimum*) at if there is some open interval containing for which for all on ; - achieves its
*absolute maximum*(or*global maximum*) value on a set if and for all in ; - achieves its
*absolute minimum*(or*global minimum*) value on a set if and for all in .

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 , for some integer ,

- achieves a
*relative maximum*(or*local maximum*) at if there is some open ball containing for which for all in ; - achieves a
*relative minimum*(or*local minimum*) at if there is some open ball containing for which for all in ; - achieves its
*absolute maximum*(or*global maximum*) value on a set if and for all in ; - achieves its
*absolute minimum*(or*global minimum*) value on a set if and for all in .

Here is a vector representing the n-tuple .

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