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 "-values") rather than what the extreme values actually are (the "-values"), whereas in the case of absolute/global extrema it is common to concentrate on the extreme value itself (the "-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..
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
- achieves a relative minimum (or local minimum) at if there is some open interval containing for which for all
- achieves its absolute maximum (or global maximum) value on a set if and for all
- achieves its absolute minimum (or global minimum) value on a set if and for all
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
- achieves a relative minimum (or local minimum) at if there is some open ball containing for which for all
- achieves its absolute maximum (or global maximum) value on a set if and for all
- achieves its absolute minimum (or global minimum) value on a set if and for all
Here is a vector representing the n-tuple .
Note that a relative/local extremum cannot happen on the boundary of the function's domain.
Finding extrema[]
Single-variable functions[]
The simplest way to find extrema of single variable functions is to take the derivative and find the stationary points, or the points at which the derivative is equal to 0 (at extrema, with the exception of endpoints on a closed interval, the slope of the tangent line is 0). The second derivative test will determine the concavity of the function at the point; if the second derivative is negative, the function will be concave down, and it will have a maximum. On a closed interval, the value of the endpoints must also be found.
Multivariable functions[]
For a multivariable function, the points to be tested are those on which all the partial derivatives are equal to 0. To determine whether a point is maximum, minimum, or saddle point, one must take every possible second derivative and construct a matrix, known as a Hessian matrix. For example, with a function of two variables, the Hessian matrix is
For three variables, this becomes
If the determinant of the Hessian positive, it will be a maximum if , , or is negative and a minimum if these second derivatives are positive. If it is negative, there will be a saddle point. If it is zero, another test must be used.