Theory of Maxima and Minima

Abstract

Publisher Summary This chapter discusses the theory of maxima and minima. The theory of ordinary maxima and minima is concerned with the problem of finding the values of each of n independent variables x 1 , x 2 , …., x n at which some function of the n variables f (x 1 , x 2 , …., x n ) reaches either a maximum or a minimum (an extremum). This problem may be interpreted geometrically as the problem of finding a point in an n-dimensional space at which the desired function has an extremum. The basic problem of the theory of ordinary maxima and minima is to determine the location of local extrema and then, compare these so as to determine which is the absolute extremum. The existence of a solution to an ordinary minimum problem is guaranteed by the theorem of Weierstrass, as long as the function is continuous. The theorem of Weierstrass indicates that the extrema may occur on the boundary of the region. There are a number of practically important problems that can be solved by using the theory of ordinary maxima and minima to optimize integrals rather than functions. A typical problem of the theory of ordinary maxima and minima would be the determination of the values of thrust and propellant weight in each stage of a multistage space vehicle that will maximize the payload for some specific mission. Several practical applications of the theory of maxima and minima are also presented in the chapter.