Maximum and minimum

2015-04-22T19:30:18Z

2001:981:F73B:1:5DB8:8779:109C:82F6: typo fixed

[[File:Extrema example original.svg|thumb|Local and global maxima and minima for cos(3π''x'')/''x'', 0.1≤''x''≤1.1]]
In [[mathematics]], the '''maximum''' and '''minimum''' of a [[function (mathematics)|function]] are the largest and smallest [[value]] that the function takes at a given point.<ref>{{cite book | last=Stewart | first=James | authorlink=James Stewart (mathematician) | title=Calculus: Early Transcendentals |publisher=[[Brooks/Cole]] | edition=6th | year=2008 | isbn=0-495-01166-5}}</ref><ref>{{cite book | last1=Larson | first1=Ron | authorlink=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=[[Brooks/Cole]] | edition=9th | year=2009 | isbn=0-547-16702-4}}</ref><ref>{{cite book | last1 = Thomas | first1 = George B. | last2=Weir | first2= Maurice D. | last3=Hass | first3=Joel |author3-link = Joel Hass| authorlink=George B. Thomas | title=Thomas' Calculus: Early Transcendentals | publisher=[[Addison-Wesley]] | year=2010 | edition=12th | isbn=0-321-58876-2}}</ref> Together they are known as the '''extrema''' (singular: extremum).

Minimum means the least you can do of something. For example, if the minimum amount of [[dollar]]s you must pay for something is seven, then you can not pay six dollars or less (you must pay at least seven). You can do more than the minimum, but no less. Maximum means the most you can have of something. For example, if the maximum amount of [[Orange (fruit)|oranges]] you can juggle is five, you can not juggle more than five oranges. You can do the maximum or less.

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]].

==References==
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[[Category:Mathematics]]

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Maximum and minimum