Extreme value
From Mathematics
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 
.
Contents |
[edit] Definitions
[edit] 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.
[edit] 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.
