Maxima and minima

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

A real-valued function f defined on a domain X has a global (or absolute) maximum point at x if f(x) ≥ f(x) for all x in X. Similarly, the function has a global (or absolute) minimum point at x if f(x) ≤ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.

If the domain X is a metric space then f is said to have a local (or relative) maximum point at the point x if there exists some ε > 0 such that f(x) ≥ f(x) for all x in X within distance ε of x. Similarly, the function has a local minimum point at x if f(x) ≤ f(x) for all x in X within distance ε of x. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods.

In both the global and local cases, the concept of a strict extremum can be defined. For example, x is a strict global maximum point if, for all x in X with xx, 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 xx, we have f(x) > f(x). Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points.

A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above).

This page was last edited on 28 April 2018, at 15:40 (UTC).
Reference: https://en.wikipedia.org/wiki/Maxima_and_minima under CC BY-SA license.

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