Advanced Calculus fi..

154 Advanced Calculus, Fifth Edition ' '* . _ ' il r 7-.one variable s, serving as coordinate on a line through (xo, yo, zo) in direction u. Toanalyze the critical point, one uses the second derivative d2w/ds2 = VUVu w:a2w a2w a2w= -cos2a+2- cos ol cos j3 + 2- cos ol cos y (2.150)ax2 ax ay ax a2If this expression is positive for all u, then w has a relative minimum at (x0, yo, zo).Algebraic criteria for the positiveness of this "quadratic form7' can be obtained; seeSection 2.21.Remarks. The tests for maxima or minima of f (x) given at the beginning of thissection can be deduced from Taylor's formula with remainder (Section 6.17). Thereis a similar formula for functions of several variables (Section 6.21), and it can alsobe used to derive the tests for maxima and minima.-4*2.20 EXTREMA FOR FUNCTIONS WITH SIDE CONDITIONSLAGRANGE MULTIPLIERSA problem of considerable importance for applications is that of maximizing orminimizing a function of several variables, where the variables are related by one ormore equations, termed side conditions. Thus the problem of finding the radius ofthe largest sphere inscribable in the ellipsoid x2 +2y2 +3z2 = 6 is equivalent to minimizingthe function w = x2 + y2 + z2, with the side condition x2 + 2y2 + 3z2 = 6.To handle such problems, one can, if possible, eliminate some of the variables byusing the side conditions and eventually reduce the problem to an ordinary maximumand minimum problem such as that considered in the preceding section. This procedureis not always feasible, and the following procedure is often more convenient; italso treats the variables in a more symmetrical way, so that various simplificationsmay be possible.To illustrate the method, we consider the problem of maximizing w = f (x, y, z),where equations g(x, y, z) = 0 and h(x, y, z) = 0 are given. The equations g = 0and h = 0 describe two surfaces in space, and the problem is thus one of maximizingf (x, y, z) as (x, y, z) varies on the curve of intersection of these surfaces. At amaximum point the derivative off along the curve, that is, the directional derivativealong the tangent to the curve, must be 0. This directional derivative is the componentof the vector V f along the tangent. It follows that V f must lie in a plane normal tothe curve at the point. This plane also contains the vectors Vg and V h (Section 2.13);that is, the vectors Vf, Vg, and Vh are coplanar at the point. Hence (Section 1.3)there must exist scalars A, and h2 such that