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In mathematical analysis, the maximum and minimum [a] of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum , [ b ] they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function.
This is a list of limits for common functions such as elementary functions. In this article, the terms a , b and c are constants with respect to x . Limits for general functions
A continuous function () on the closed interval [,] showing the absolute max (red) and the absolute min (blue).. In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval [,], then must attain a maximum and a minimum, each at least once.
After establishing the critical points of a function, the second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. [1] If the function f is twice-differentiable at a critical point x (i.e. a point where f ′ (x) = 0), then:
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. It is attributed to the German mathematician Heinz Bauer. [1]
The disks are laid such that their centers form a polygonal path from the value where () is maximized to any other point in the domain, while being totally contained within the domain. Thus the existence of a maximum value implies that all the values in the domain are the same, thus () is constant.
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 weak maximum principle, in this setting, says that for any open precompact subset M of the domain of u, the maximum of u on the closure of M is achieved on the boundary of M. The strong maximum principle says that, unless u is a constant function, the maximum cannot also be achieved anywhere on M itself.