Maximum

The largest value of a set, function, etc. The maximum value of a set of elements is denoted or , and is equal to the last element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the maximum is 5. The maximum and minimum are the simplest order statistics.

The maximum value of a variable is commonly denoted (Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 74). In this work, the convention is used.

The maximum of a set of elements is implemented in the Wolfram Language as Max[list] and satisfies the identities

			(1)
			(2)

Definite integrals include

			(3)
			(4)

A continuous function may assume a maximum at a single point or may have maxima at a number of points. A global maximum of a function is the largest value in the entire range of the function, and a local maximum is the largest value in some local neighborhood.

For a function which is continuous at a point , a necessary but not sufficient condition for to have a local maximum at is that be a critical point (i.e., is either not differentiable at or is a stationary point, in which case ).

The first derivative test can be applied to continuous functions to distinguish maxima from minima. For twice differentiable functions of one variable, , or of two variables, , the second derivative test can sometimes also identify the nature of an extremum. For a function , the extremum test succeeds under more general conditions than the second derivative test.