Extreme value: Difference between revisions
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"Minimum" and "Maximum" redirect here. For minimum or maximum of a function, see [[maxima and minima]]. | |||
For a [[differentiable]] [[function (mathematics)|function]] ''f'', if ''f''(''x''<sub>0</sub>) is an extreme value for the set of all values ''f''(''x''), and if ''f''(''x''<sub>0</sub>) is in the [[interior]] of the [[domain (mathematics)|domain]] of ''f'', then ''x''<sub>0</sub> is a [[Critical_point_(mathematics)|critical point]]. | The "largest" and the "smallest" element of a [[set]] are called '''extreme values'''. In general we need to distinguish various possible meanings of "largest" and "smallest" | ||
==Linear order== | |||
In a [[linear order|linearly ordered]] set, such as the [[real number]]s, any two elements ''x'' and ''y'' are comparable. We suppose for definiteness that <math>x \ge y</math>, and so we may define the '''maximum''' of the set {''x'',''y''} to be ''x'' and the '''minimum''' to be ''y''. By [[iteration]] we may define the maximum and minimum of any (non-empty) finite set. | |||
Let ''S'' be a subset of a linearly ordered set (''X'',<). An ''upper bound'' for ''S'' is an element ''U'' of ''X'' such that <math>U \ge s</math> for all elements <math>s \in S</math>. A ''lower bound'' for ''S'' is an element ''L'' of ''X'' such that <math>L \le s</math> for all elements <math>s \in S</math>. A set is ''bounded'' if it has both lower and upper bounds. In general a set need not have either an upper or a lower bound. | |||
A ''supremum'' for ''S'' is an upper bound which is less than or equal to any other upper bound for ''S''; an ''infimum'' is a lower bound for ''S'' which is greater than or equal to any other lower bound for ''S''. In general a set with upper bounds need not have a supremum; a set with lower bounds need not have an infimum. The supremum or infimum of ''S'', if one exists, is unique. | |||
A ''maximum'' for ''S'' is an upper bound which is in ''S''; a ''minimum'' for ''S'' is a lower bound which is in ''S''. A maximum is necessarily a supremum, but a supremum for a set need not be a maximum (that is, need not be an element of the set); similarly an infimum need not be a minimum. As noted above, any finite set has a maximum and minimum which are thus its supremum and infimum. | |||
The fundamental axiom for the real numbers is that every non-empty bounded set has a supremum and an infimum. | |||
===Algebraic properties=== | |||
Maximum and minimum are [[binary operation]]s on a linearly ordered set, sometimes written <math>x \vee y</math> and <math>x \wedge y</math> respectively, satisfying the following properties: | |||
* [[Idempotence]]: <math>x \vee x = x = x \wedge x ; \,</math> | |||
* [[Commutativity]]: <math>x \vee y = y \vee x;~ x \wedge y = y \wedge x ;\,</math> | |||
* [[Associativity]]: <math>x \vee (y \vee z) = (x \vee y) \vee z;~ x \wedge (y \wedge z) = (x \wedge y) \wedge z ;\,</math> | |||
* [[Absorption (mathematics)]]: <math>x \wedge (x \vee y) = x;~ x \vee (x \wedge y) = x.\,</math> | |||
These are characterising properties of a [[lattice (order)|lattice]]. | |||
==Critical points== | |||
For a [[differentiable]] [[function (mathematics)|function]] ''f'', if ''f''(''x''<sub>0</sub>) is an extreme value for the set of all values ''f''(''x''), and if ''f''(''x''<sub>0</sub>) is in the [[interior (topology)|interior]] of the [[domain (mathematics)|domain]] of ''f'', then ''x''<sub>0</sub> is a [[Critical_point_(mathematics)|critical point]]. | |||
== See also == | == See also == | ||
*[[Minima and maxima]] | *[[Minima and maxima]] | ||
Latest revision as of 18:17, 17 January 2010
"Minimum" and "Maximum" redirect here. For minimum or maximum of a function, see maxima and minima.
The "largest" and the "smallest" element of a set are called extreme values. In general we need to distinguish various possible meanings of "largest" and "smallest"
Linear order
In a linearly ordered set, such as the real numbers, any two elements x and y are comparable. We suppose for definiteness that , and so we may define the maximum of the set {x,y} to be x and the minimum to be y. By iteration we may define the maximum and minimum of any (non-empty) finite set.
Let S be a subset of a linearly ordered set (X,<). An upper bound for S is an element U of X such that for all elements . A lower bound for S is an element L of X such that for all elements . A set is bounded if it has both lower and upper bounds. In general a set need not have either an upper or a lower bound.
A supremum for S is an upper bound which is less than or equal to any other upper bound for S; an infimum is a lower bound for S which is greater than or equal to any other lower bound for S. In general a set with upper bounds need not have a supremum; a set with lower bounds need not have an infimum. The supremum or infimum of S, if one exists, is unique.
A maximum for S is an upper bound which is in S; a minimum for S is a lower bound which is in S. A maximum is necessarily a supremum, but a supremum for a set need not be a maximum (that is, need not be an element of the set); similarly an infimum need not be a minimum. As noted above, any finite set has a maximum and minimum which are thus its supremum and infimum.
The fundamental axiom for the real numbers is that every non-empty bounded set has a supremum and an infimum.
Algebraic properties
Maximum and minimum are binary operations on a linearly ordered set, sometimes written and respectively, satisfying the following properties:
These are characterising properties of a lattice.
Critical points
For a differentiable function f, if f(x0) is an extreme value for the set of all values f(x), and if f(x0) is in the interior of the domain of f, then x0 is a critical point.