The smallest value of a set, function, etc. The minimum value of a set of elements is denoted or ,
and is equal to the first element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the minimum is 1. The maximum
and minimum are the simplest order statistics.
The minimum value of a variable is commonly denoted (cf. Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 84).
In this work, the convention
is used.
The minimum of a set of elements is implemented in the Wolfram Language as Min[list]
and satisfies the identities
(1)
(2)
A continuous function may assume a minimum at a single point or may have minima at a number of points. A global
minimum of a function is the smallest value in the
entire range of the function,
while a local minimum is the smallest value in some
local neighborhood.
is denoted 
or 
,
and is equal to the first element of a sorted (i.e., ordered) version of 
. For example, given the set 
, the sorted version is 
, so the minimum is 1. The maximum
and minimum are the simplest order statistics.
is commonly denoted 
(cf. Strang 1988, pp. 286-287 and 301-303) or 
(Golub and Van Loan 1996, p. 84).
In this work, the convention 
is used.
which is continuous at a point 
, a necessary but not sufficient
condition for 
to have a local minimum at 
is that 
be a critical point (i.e.,
is either not differentiable
at 
or 
is a stationary point,
in which case 
).
,
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.
