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单词 Extremum
释义

extremum


Extrema are minima and maxima. The forms of these are extremum, minimum, and maximum.

Extrema may be “global” or “local”. A global minimum of a functionMathworldPlanetmath f is the lowest value that f ever achieves. If you imagine the function as a surface, then a global minimum is the lowest point on that surface. Formally, it is said that f:UV has a global minimum at x if uU,f(x)f(u).

A local minimum of a function f is a point x which has less value than all points “next to” it. If you imagine the function as a surface, then a local minimum is the of a “valley” or “bowl” in the surface somewhere. Formally, it is said that f:UV has a local minimum at x if a neighborhoodMathworldPlanetmath N of x such that yN, f(x)f(y).

If you flip the signs above to , you get the definitions of global and local maxima.

A “strict local minima” or “strict local maxima” means that nearby points are strictly less than or strictly greater than the critical pointDlmfMathworld, rather than or . For instance, a strict local minima at x has a neighborhood N such that yN, (f(x)<f(y) or y=x).

A saddle point is a critical point which is not a local extremum.

A related concept is plateau.

Finding minima or maxima is an important task which is part of the of optimization.This task is also important in Physics where the minima correspond to equilibria.

Titleextremum
Canonical nameExtremum
Date of creation2013-03-22 12:59:44
Last modified on2013-03-22 12:59:44
Ownerbshanks (153)
Last modified bybshanks (153)
Numerical id20
Authorbshanks (153)
Entry typeDefinition
Classificationmsc 26B12
Synonymextrema
Related topicPlateau
Related topicRelationsBetweenHessianMatrixAndLocalExtrema
Related topicLeastAndGreatestValueOfFunction
Related topicSpeediestInclinedPlane
Definesglobal minima
Definesglobal maxima
Defineslocal minima
Defineslocal maxima
Definesglobal minimum
Definesglobal maximum
Defineslocal minimum
Defineslocal maximum
Definesstrict local minima
Definesstrict local maxima
Definesstrict local minimum
Definesstrict local maximum
Definessaddle point
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更新时间:2025/5/4 17:03:31