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.

Title extremum
Canonical name Extremum
Date of creation 2013-03-22 12:59:44
Last modified on 2013-03-22 12:59:44
Owner bshanks (153)
Last modified by bshanks (153)
Numerical id 20
Author bshanks (153)
Entry type Definition
Classification msc 26B12
Synonym extrema
Related topic Plateau
Related topic RelationsBetweenHessianMatrixAndLocalExtrema
Related topic LeastAndGreatestValueOfFunction
Related topic SpeediestInclinedPlane
Defines global minima
Defines global maxima
Defines local minima
Defines local maxima
Defines global minimum
Defines global maximum
Defines local minimum
Defines local maximum
Defines strict local minima
Defines strict local maxima
Defines strict local minimum
Defines strict local maximum
Defines saddle point