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An absolute extremum (or global extremum) of a function in a given interval is the point at which a maximum or minimum value of the function is obtained. Frequently, the interval given is the function's domain, and the absolute extremum is the point corresponding to the maximum or minimum value of the entire function.
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Definition. Throughout, it is assumed that is a real or complex vector space.. For any ,,, say that lies between and if and there exists a < < such that = + ().. If is a subset of and , then is called an extreme point of if it does not lie between any two distinct points of . That is, if there does not exist , and < < such that and = + (). The set of all extreme points of is …
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If a function f(x) is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b]. If f(x) has an extremum on an open interval (a,b), then the extremum occurs at a critical point. This theorem is sometimes also called the Weierstrass extreme value theorem. The standard proof of the first …
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