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In mathematical analysis, the maxima and minima (the plural 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. ==Definition== 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. Note that a global maximum point is always a local maximum point, and similarly for minimum points. 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 ''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 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「maxima and minima」の詳細全文を読む スポンサード リンク
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