Extreme value: Difference between revisions
Content deleted Content added
roll back. i came here looking for the maximum of a function. the link should stay. |
m Bot: Fixing double redirect to Maximum and minimum Tag: Redirect target changed |
||
| (22 intermediate revisions by 17 users not shown) | |||
| Line 1: | Line 1: | ||
#REDIRECT [[Maximum and minimum]] |
|||
{{redirect|Maximum|maximum of a function|maxima and minima}} |
|||
{{R from merge}} |
|||
The largest and the smallest element of a [[set]] are called '''extreme values''', '''absolute extrema''', or '''extreme records'''. |
|||
{{R hatnote}} |
|||
For a [[differentiable]] [[function (mathematics)|function]] <math>f</math>, if <math>f(x_0)</math> is an extreme value for the set of all values <math>f(x)</math>, and if <math>x_0</math> is in the [[interior]] of the [[domain (mathematics)|domain]] of <math>f</math>, then <math>x_0</math> is a [[Critical_point_(mathematics)|critical point]], by [[Fermat's theorem (stationary points)|Fermat's theorem]]. |
|||
== Extreme values in abstract spaces with order == |
|||
In the case of a general [[partial order]] one should not confuse a '''least element''' (smaller than all other) and a '''minimal element''' (nothing is smaller). Likewise, a '''[[greatest element]]''' of a [[poset]] is an [[upper bound]] of the set which is contained within the set, whereas a '''maximal element''' ''m'' of a poset ''A'' is an element of ''A'' such that if ''m'' ≤ ''b'' (for any ''b'' in ''A'') then ''m'' = ''b''. |
|||
Any least element or greatest element of a poset will be unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable. |
|||
In a [[total order|totally ordered]] set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element and the maximal element will also be the greatest element. Thus in a totally ordered set we can simply use the terms '''''minimum''''' and '''''maximum'''''. |
|||
If a chain is finite then it will always have a maximum and a minimum. If a chain is infinite then it need not have a maximum or a minimum. For example, the set of [[natural number]]s has no maximum, though it has a minimum. |
|||
If an infinite chain ''S'' is bounded, then the [[topological closure|closure]] ''Cl(S)'' of the set occasionally has a minimum and a maximum, in such case they are called the '''[[infimum|greatest lower bound]]''' and the '''[[supremum|least upper bound]]''' of the set ''S'', respectively. |
|||
In general, if an ordered set ''S'' has a greatest element m, m is a maximal element. Furthermore, if ''S'' is a subset of an ordered set ''T'' and m is the greatest element of ''S'' with respect to order induced by ''T'', m is a least upper bound of ''S'' in ''T''. The similar result holds for least element, minimal element and greatest lower bound. |
|||
==See also== |
|||
{{Wiktionarypar|minimum|maximum}} |
|||
* [[Extreme point]] |
|||
* [[Extreme value theorem]] |
|||
* [[Extreme value theory]] |
|||
* [[Fermat's theorem (stationary points)|Fermat's theorem]] |
|||
* [[Generalized extreme value distribution]] |
|||
[[Category:Calculus]] |
|||
[[Category:Order theory]] |
|||
{{Link FA|pl}} |
|||
[[ar:حدود عليا وحدود دنيا]] |
|||
[[de:Extremwert]] |
|||
[[fr:Extremum]] |
|||
[[is:Útgildi]] |
|||
[[nl:Extreme waarden]] |
|||
[[pl:Ekstremum]] |
|||
[[pt:Mínimo]] |
|||
[[fi:Maksimi]] |
|||
Latest revision as of 00:34, 12 February 2023
Redirect to:
- From a merge: This is a redirect from a page that was merged into another page. This redirect was kept in order to preserve the edit history of this page after its content was merged into the content of the target page. Please do not remove the tag that generates this text (unless the need to recreate content on this page has been demonstrated) or delete this page.
- For redirects with substantive page histories that did not result from page merges use {{R with history}} instead.
- Mentioned in a hatnote: This is a redirect from a title that is mentioned in a hatnote at the redirect target. The mention is usually atop the target article. It may, however, be directly under a section header, or in another article's hatnote (whenever the hatnote is under a section, {{R to section}} should also be used).
- The titles of redirects mentioned in hatnotes may refer to a subject other than that of the target page. It is possible that this redirect may need to be retargeted, or become an article under its own title (see {{R with possibilities}}). If the title is a good candidate for a Wiktionary link, it may also be added.