Minima and maxima: Difference between revisions
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In [[mathematics]], '''minima''' and '''maxima''', known collectively as '''extrema''', are the or ''smallest value'' (minimum) ''largest value'' (maximuml), 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). | In [[mathematics]], '''minima''' and '''maxima''', known collectively as '''extrema''', are the or ''smallest value'' (minimum) ''largest value'' (maximuml), 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 == | |||
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'') when |''x'' − ''x''<sup>*</sup>| < ε. The value of the function at this point is called '''minimum''' of the function. | |||
== See also == | == See also == | ||
*[[Extreme value]] | *[[Extreme value]] | ||
Revision as of 23:14, 23 November 2007
In mathematics, minima and maxima, known collectively as extrema, are the or smallest value (minimum) largest value (maximuml), that a function takes in a point either within a given neighbourhood (local extremum) or on the whole function domain (global extremum).
Definition
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) when |x − x*| < ε. The value of the function at this point is called minimum of the function.