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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.

See alsoEdit

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