Maximum and minimum points

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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 is defined on a topological space , then a point is called a local maximum (local minimum) point if there is a neighbourhood of such that is an absolute maximum (minimum) point for the restriction of to this neighbourhood. One distinguishes between strict and non-strict maximum (minimum) points (both absolute and local). For example, a point is called a non-strict (strict) local maximum point of if there is a neighbourhood of such that for all , (, ).

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 be defined in a neighbourhood of a point of the real line. If is a non-strict local maximum (minimum) point and if the derivative exists, then the latter is equal to zero.

If a function is differentiable in a neighbourhood of except, possibly, at itself where it is continuous, and if the derivative is of constant sign on each side of in this neighbourhood, then for 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, for and for (respectively, from minus to plus; for and for ). However, it is not possible to speak of the change of sign of the derivative at for every function that is differentiable in a neighbourhood of .

If has derivatives at and if , , , then for to be a strict local maximum point it is necessary and sufficient that be even and that , and for a local minimum that be even and .

Let be defined in an -dimensional neighbourhood of a point and let it be differentiable at this point. If is a non-strict local maximum (minimum) point, then the differential of at this point is equal to zero. This condition is equivalent to all first-order partial derivatives of being zero at this point. If the function has continuous second-order partial derivatives at , if all its first-order derivatives are equal to zero at , and if the second-order differential at is a negative-definite (positive-definite) quadratic form, then 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 and mathematical programming (see also Maximization and minimization of functions). Maximum and minimum points of functions on manifolds are studied in variational calculus in the large, and maximum and minimum points for functions on function spaces, that is, for functionals, are studied in variational calculus. There are also various numerical approximation methods for finding maximum and minimum points.


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



[a1] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Maximum and minimum points. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098