Minima and maxima: Difference between revisions
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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 == | ||
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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]][[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:01, 19 September 2024
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 ≤.