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Points in the domain of definition of a real-valued function at which it takes its greatest and smallest values; such points are also called absolute maximum and absolute minimum points. If  $  f $
is defined on a topological space  $  X $, 
then a point  $  x _ {0} $
is called a local maximum (local minimum) point if there is a neighbourhood  $  U \subseteq X $
of  $  x _ {0} $
such that  $  x _ {0} $
is an absolute maximum (minimum) point for the restriction of  $  f $
to this neighbourhood. One distinguishes between strict and non-strict maximum (minimum) points (both absolute and local). For example, a point  $  x _ {0} \in \mathbf R $
is called a non-strict (strict) local maximum point of  $  f $
if there is a neighbourhood  $  U $
of  $  x _ {0} $
such that for all  $  x \in U $, 
$  f ( x) \leq  f ( x _ {0} ) $(
$  f( x) < f ( x _ {0} ) $, 
$  x \neq x _ {0} $).

For functions defined on finite-dimensional domains there are conditions and tests, in terms of differential calculus, for a given point to be a local maximum (minimum) point. Let  $  f $
be defined in a neighbourhood of a point  $  x _ {0} $
of the real line. If  $  x _ {0} $
is a non-strict local maximum (minimum) point and if the derivative  $  f ^ { \prime } ( x _ {0} ) $
exists, then the latter is equal to zero.

If a function  $  f $
is differentiable in a neighbourhood of  $  x _ {0} $
except, possibly, at  $  x _ {0} $
itself where it is continuous, and if the derivative  $  f ^ { \prime } $
is of constant sign on each side of  $  x _ {0} $
in this neighbourhood, then for  $  x _ {0} $
to be a strict local maximum (local minimum) point it is necessary and sufficient that the derivative changes sign from plus to minus, that is,  $  f ^ { \prime } ( x) > 0 $
for  $  x < x _ {0} $
and  $  f ^ { \prime } ( x) < 0 $
for  $  x > x _ {0} $(
respectively, from minus to plus;  $  f ^ { \prime } ( x) < 0 $
for  $  x < x _ {0} $
and  $  f ^ { \prime } ( x) > 0 $
for  $  x > x _ {0} $). 
However, it is not possible to speak of the change of sign of the derivative at  $  x _ {0} $
for every function  $  f $
that is differentiable in a neighbourhood of  $  x _ {0} $.

If  $  f $
has  $  m $
derivatives at  $  x _ {0} $
and if  $  f ^ { ( k) } ( x _ {0} ) = 0 $, 
$  k = 1 \dots m - 1 $, 
$  f ^ { ( m) } ( x _ {0} ) \neq 0 $, 
then for  $  x _ {0} $
to be a strict local maximum point it is necessary and sufficient that  $  m $
be even and that  $  f ^ { ( m) } ( x _ {0} ) < 0 $, 
and for a local minimum that  $  m $
be even and  $  f ^ { ( m) } ( x _ {0} ) > 0 $.

Let  $  f ( x _ {1} \dots x _ {n} ) $
be defined in an  $  n $-
dimensional neighbourhood of a point  $  x  ^ {(} 0) = ( x _ {1}  ^ {(} 0) \dots x _ {n}  ^ {(} 0) ) $
and let it be differentiable at this point. If  $  x  ^ {(} 0) $
is a non-strict local maximum (minimum) point, then the differential of  $  f $
at this point is equal to zero. This condition is equivalent to all first-order partial derivatives of  $  f $
being zero at this point. If the function has continuous second-order partial derivatives at  $  x  ^ {(} 0) $, 
if all its first-order derivatives are equal to zero at  $  x  ^ {(} 0) $, 
and if the second-order differential at  $  x  ^ {(} 0) $
is a negative-definite (positive-definite) quadratic form, then  $  x  ^ {(} 0) $
is a strict local maximum (minimum) point. Conditions for maximum and minimum points of differentiable functions are known when restrictions are imposed on the variation of the arguments in the domain: coupling equations must be satisfied. Necessary and sufficient conditions for a maximum (minimum) of a real-valued function with a more complicated structure of its domain of definition have been obtained in special areas of mathematics; for example, in [[Convex analysis|convex analysis]] and [[Mathematical programming|mathematical programming]] (see also [[Maximization and minimization of functions|Maximization and minimization of functions]]). Maximum and minimum points of functions on manifolds are studied in [[Variational calculus in the large|variational calculus in the large]], and maximum and minimum points for functions on function spaces, that is, for functionals, are studied in [[Variational calculus|variational calculus]]. There are also various numerical approximation methods for finding maximum and minimum points.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1''' , MIR  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,   "A course in mathematical analysis" , '''1–2''' , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>