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A point x0 ; y0 is called a relative maximum point or relative minimum point of f x; y respectively according as f x0 h; y0 k < f x0 ; y0 or f x0 h; y0 k > f x0 ; y0 for all h and k such that 0 < jhj < ; 0 < jkj <  where  is a su ciently small positive number. A necessary condition that a di erentiable function f x; y have a relative maximum or minimum is @f 0; @x @f 0 @y 19
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If x0 ; y0 is a point (called a critical point) satisfying equations (19) and if is de ned by 8 ! ! !2 9 < @2 f = @2 f @2 f   : @x2 @y2 @x @y ;
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then 1. x0 ; y0 is a relative maximum point if > 0 and  @2 f   <0 @x2  x0 ;y0  @2 f   x0 ; y0 is a relative minimum point if > 0 and >0 @x2  x0 ;y0 or
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!  @2 f   <0 @y2  x0 ;y0 !  @2 f   >0 @y2  x0 ;y0
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x0 ; y0 is neither a relative maximum or minimum point if < 0. If < 0, x0 ; y0 is sometimes called a saddle point. No information is obtained if 0 (in such case further investigation is necessary).
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METHOD OF LAGRANGE MULTIPLIERS FOR MAXIMA AND MINIMA A method for obtaining the relative maximum or minimum values of a function F x; y; z subject to a constraint condition  x; y; z 0, consists of the formation of the auxiliary function G x; y; z  F x; y; z  x; y; z subject to the conditions @G 0; @x @G 0; @y @G 0 @z 22 21
which are necessary conditions for a relative maximum or minimum. The parameter , which is independent of x; y; z, is called a Lagrange multiplier. The conditions (22) are equivalent to rG 0, and hence, 0 rF r Geometrically, this means that rF and r are parallel. This fact gives rise to the method of Lagrange multipliers in the following way. Let the maximum value of F on  x; y; z 0 be A and suppose it occurs at P0 x0 ; y0 ; z0 . (A similar argument can be made for a minimum value of F.) Now consider a family of surfaces F x; y; z C. The member F x; y; z A passes through P0 , while those surfaces F x; y; z B with B < A do not. (This choice of a surface, i.e., f x; y; z A, geometrically imposes the condition  x; y; z 0 on F.) Since at P0 the condition 0 rF r tells us that the gradients of F x; y; z A and  x; y; z are parallel, we know that the surfaces have a common tangent plane at a point that is maximum for F. Thus, rG 0 is a necessary condition for a relative maximum of F at P0 . Of course, the condition is not su cient. The critical point so determined may not be unique and it may not produce a relative extremum. The method can be generalized. If we wish to nd the relative maximum or minimum values of a function F x1 ; x2 ; x3 ; . . . ; xn subject to the constraint conditions  x1 ; . . . ; xn 0; 2 x1 ; . . . ; xn 0; . . . ; k x1 ; . . . ; xn 0, we form the auxiliary function
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