Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 147This function has the absolute minimum MI = 0 and the absolute maximum M2 = 1and is bounded, with K = 1. The example y = x for 0 < x < 1 illustrates a boundedfunction (K = 1) having neither absolute minimum nor absolute maximum; theexample y = tanx for -in < x < in illustrates an unbounded function.To find the absolute maximum M of a function y = f (x), differentiable fora 5 x 5 b, one can reason as follows: if f (xo) = M and a < xo < b, then f (x) hasnecessarily a relative maximum at xo; thus the absolute maximum occurs either at acritical point within the interval or at x = a or x = b. One therefore first locates allcritical points within the interval and compares the values of y at these points withthe values at x = a and x = b; the largest value of y is the maximum sought. It isthus not necessary to use the second derivative test described at the beginning of thissection. The absolute minimum can be found in the same way.The determination of critical points and their classification into maxima, minima,or neither are important also for graphing functions; from a knowledge of thecritical points and the corresponding values of y, a very good first approximation tothe graph of y = f (x) can be obtained.After these preliminaries we can consider the analogous questions for functionsof two or more variables. Let z = f (x, y) be defined and continuous in a domain D.This function is said to have a relative maximum at (xo, yo) if f (x, y) I f (xo, yo)for (x, y) sufficiently close to (xo, yo) and to have a relative minimum at (xO, yo) iff (x, y) >_ f (xo, yo) for (x, y) sufficiently close to (xo, yo). Let (xo, yo) give a relativemaximum of f (x, y); then the function f (x, yo), which depends on x alone, has arelative maximum at xo, as is illustrated in Fig. 2.16. Hence if fx(xo, yo) exists,then fx(xo, yo) = 0; similarly, if f,(xo, yo) exists, then fy(xo, yo) = 0. Points (x, y)at which both partial derivatives are 0 are termed critical points off. As before,one concludes that every relative maximum and every relative minimum occur at acritical point off, if f, and f, exist in D.One might expect that the nature of a critical point could be determined bystudying the functions f (x, yo) and f (xa, y), with the aid of the second derivativesas previously. First of all, it should be remarked that one of these functions can havea maximum at (xO, yo), while the other has a minimum. This is illustrated by thefunction z = 1 + x2 - y2 at (0,O); th~s function has a minimum with respect to x atx = 0 for y = 0 and a maximum with respect to y at y = 0 for x = 0, as shown inI I I /'YOI I I/----- Y -----A1-YXFigure 2.16 Maximum of z = f (x, y).