Minima and maxima: Difference between revisions
Jump to navigation
Jump to search
imported>Igor Grešovnik No edit summary |
imported>Jitse Niesen m (spelling) |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | {{subpages}} | ||
In [[mathematics]], '''minima''' and '''maxima''', known collectively as '''extrema''', are the | In [[mathematics]], '''minima''' and '''maxima''', known collectively as '''extrema''', are the ''smallest value'' (minimum) or ''largest value'' (maximum), that a [[function (mathematics)|function]] takes in a point either within a given neighbourhood (local extremum) or on the whole function [[domain (mathematics)|domain]] (global extremum). | ||
== Definition == | == Definition == | ||
=== Minimum === | |||
A real-valued [[function (mathematics)|function]] ''f'' is said to have a '''local minimum''' at the point ''x''<sup>*</sup>, if there exists some ε > 0, such that ''f''(''x''<sup>*</sup>) ≤ ''f''(''x'') whenever |''x'' − ''x''<sup>*</sup>| < ε. The value of the function at this point is called '''minimum''' of the function. | A real-valued [[function (mathematics)|function]] ''f'' is said to have a '''local minimum''' at the point ''x''<sup>*</sup>, if there exists some ε > 0, such that ''f''(''x''<sup>*</sup>) ≤ ''f''(''x'') whenever |''x'' − ''x''<sup>*</sup>| < ε. The value of the function at this point is called '''minimum''' of the function. | ||
The definition of a '''local maximum''' is similar, only with the ≥ sign in place of ≤. | |||
== See also == | == See also == | ||
*[[Extreme value]] | *[[Extreme value]] | ||
Latest revision as of 11:38, 18 January 2008
In mathematics, minima and maxima, known collectively as extrema, are the smallest value (minimum) or largest value (maximum), that a function takes in a point either within a given neighbourhood (local extremum) or on the whole function domain (global extremum).
Definition
Minimum
A real-valued function f is said to have a local minimum at the point x*, if there exists some ε > 0, such that f(x*) ≤ f(x) whenever |x − x*| < ε. The value of the function at this point is called minimum of the function.
The definition of a local maximum is similar, only with the ≥ sign in place of ≤.