Conditional Relative Extremum

A relative extremum of the function f(x₁,..., x_n+m) of n + m variables is said to be conditional if the variables are subject to the m conditions

(*) φ_k(X₁..., X_n+m) = 0 1 ≤ k ≤ m

More precisely, the function f has a conditional relative extre-mum at a point M, whose coordinates satisfy equations (*) if the function’s value at M is a maximum or minimum relative to the values of f at the points in some neighborhood of M whose coordinates satisfy equations (*).

Geometrically, in the simplest case the conditional relative extremum of the function f(x, y) under the condition φ(x, y) = 0 is the highest or lowest (relative to nearby points) point on a curve that lies on the surface z = f(x, y) and whose projection on the x y plane is the curve φ(x, y) = 0. At a conditional relative extremum point the curve φ(x, y) = 0 either has a singular point or is tangent to the corresponding level line of the function f(x, y). If certain additional requirements are imposed on equations (*), the problem of finding a conditional relative extremum of I can be reduced to that of finding an ordinary relative extremum of the function: the variables x_n+1,. . ., x_n+m from equations (*) are expressed in terms of X₁, . . ., x_n, and the resulting expressions are substituted into f. Another way of solving the problem is provided by Lagrange’s method of multipliers.

Problems involving conditional extrema arise not only in geometry, where, for example, it may be desired to find the rectangle with the smallest perimeter for a given area, but also in such fields as mechanics and economics.

Many problems of the calculus of variations involve finding extrema of functionals under the condition that other functions have a given value (seeISOPERIMETRIC PROBLEMS) or finding the extremum of a functional in a class of functions that satisfy some conditions. Such problems are also solved by Lagrange’s method of multipliers. (See alsoLINEAR PROGRAMMING and MATHEMATICAL PROGRAMMING.)