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For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗, we have f(x ∗) > f(x), and x ∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we have f(x ∗) > f(x). Note that a point is a strict global maximum point if and only if ...
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value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.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.
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.
In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations. [1] In a simple two-dimensional case, consider a function of two variables u(x,y) such that
Figure 1: The red curve shows the constraint g(x, y) = c. The blue curves are contours of f(x, y). The point where the red constraint tangentially touches a blue contour is the maximum of f(x, y) along the constraint, since d 1 > d 2.
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In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. [1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions ...
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