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Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the ...
For an ordered space (X, <) (i.e. a totally ordered set equipped with the order topology), the following are equivalent: (X, <) is compact. Every subset of X has a supremum (i.e. a least upper bound) in X. Every subset of X has an infimum (i.e. a greatest lower bound) in X. Every nonempty closed subset of X has a maximum and a minimum element.
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain. [ 2 ] [ 3 ] Properties of a single space
The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of , and the minimum of is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.
Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
The discrete Chebyshev distance between two spaces on a chessboard gives the minimum number of moves a king requires to move between them. This is because a king can move diagonally, so that the jumps to cover the smaller distance parallel to a row or column is effectively absorbed into the jumps covering the larger.
Bauer's maximum principle is the following theorem in mathematical optimization: Any function that is convex and continuous , and defined on a set that is convex and compact , attains its maximum at some extreme point of that set.