For use in statistics, see Sample maximum and minimum and Extreme value theory. Maximum” and “Minimum” redirect here. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, maxima and minima problems pdf 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. 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. 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. 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. Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem global maxima and minima exist.
Local extrema of differentiable functions can be found by Fermat’s theorem, which states that they must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability. The function x3 has no global minima or maxima.