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.
