Advanced Calculus fi..

150 Advanced Calculus, Fifth EditionFigure 2.20Critical point of z = f (x, y) analyzed by vertical sections.This is precisely the quantity a2z/ar2 of the preceding example. In order to guaranteea relative maximum for z at (xo, yo), one can require that this second derivative benegative for all a, 0 5 a 5 2n; this ensures that z has a relative maximum as afunction of s in each plane in direction a, and in fact (all second derivatives beingassumed continuous) that z has a relative maximum at the point; cf. Problem 10following Section 2.2 1. A similar reasoning applies to relative minima. One thereforeobtains the rule:If azlax = 0 and azlay = 0 at (.xO, yo) andfor x = xo, y = yo, and all a: 0 5 a 5 2n, then z = f (x, y) has a relative maximumat (xo, yo); if azlax = 0 and az/ay = 0 anda2z- c0s2, + 2- in a cos u + 7 a2z sin a s 0ax2 ax ay ay-forx = xo, y = yo, and all a: 0 5 a 5 27r, then z = f (x, y) has a relative minimumat (xo, YO).It thus appears that the study of the critical points is reduced to the analysis ofthe expressionwhere the abbreviationsa2z a2z a2zA = l(xo, yo), B = -(xo, YO), C = -(xo,ax ax ay ay2yo) (2.145)are used. Here an algebraic analysis reduces the question to a simpler one, for onehas the following theorem:THEOREM If B~ - AC t 0 and A + C < 0, then the expression (2.144) is negativefor all a; if g2 - AC < 0 and A + C > 0, then the expression (2.144) is positivefor all a.I