Page 197 - Schaum's Outline of Theory and Problems of Advanced Calculus

188 APPLICATIONS OF PARTIAL DERIVATIVES [CHAP. 8

A point ðx 0 ; y 0 Þ is called a relative maximum point or relative minimum point of f ðx; yÞ respectively
according as f ðx 0 þ h; y 0 þ kÞ < f ðx 0 ; y 0 Þ or f ðx 0 þ h; y 0 þ kÞ > f ðx 0 ; y 0 Þ for all h and k such that
0 < jhj < ; 0 < jkj < where is a sufficiently small positive number.
A necessary condition that a differentiable function f ðx; yÞ have a relative maximum or minimum is
@ f @ f
¼ 0; ¼ 0
@x @y ð19Þ
If ðx 0 ; y 0 Þ is a point (called a critical point) satisfying equations (19) and if is defined by
8 9
! ! ! 2
@ f @ f @ f
< 2 2 2 =

¼ 2 2 ð20Þ
@x @y @x @y
: ;
ðx 0 ;y 0 Þ
then
!
2 2
1. ðx 0 ; y 0 Þ is a relative maximum point if > 0 and @ f < 0 or @ f < 0
@x 2 @y 2
ðx 0 ;y 0 Þ ðx 0 ;y 0 Þ
!
2 2
2. ðx 0 ; y 0 Þ is a relative minimum point if > 0 and @ f > 0 or @ f > 0
@x 2 @y 2
ðx 0 ;y 0 Þ ðx 0 ;y 0 Þ
3. ðx 0 ; y 0 Þ is neither a relative maximum or minimum point if < 0. If < 0, ðx 0 ; y 0 Þ is some-
times called a saddle point.
4. No information is obtained if ¼ 0 (in such case further investigation is necessary).
METHOD OF LAGRANGE MULTIPLIERS FOR MAXIMA AND MINIMA
A method for obtaining the relative maximum or minimum values of a function Fðx; y; zÞ subject to
a constraint condition ðx; y; zÞ¼ 0, consists of the formation of the auxiliary function

Gðx; y; zÞ Fðx; y; zÞþ ðx; y; zÞ ð21Þ
subject to the conditions
@G @G @G
¼ 0; ¼ 0; ¼ 0
@x @y @z ð22Þ
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 ¼ 0, and hence, 0 ¼rF þ 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 ðx; y; zÞ¼ 0be A and suppose it occurs at P 0 ðx 0 ; y 0 ; z 0 Þ. (A
similar argument can be made for a minimum value of F.) Now consider a family of surfaces
Fðx; y; zÞ¼ C.
The member Fðx; y; zÞ¼ A passes through P 0 , while those surfaces Fðx; y; zÞ¼ B with B < A do
not. (This choice of a surface, i.e., f ðx; y; zÞ¼ A, geometrically imposes the condition ðx; y; zÞ¼ 0on
F.) Since at P 0 the condition 0 ¼rF þ r tells us that the gradients of Fðx; y; zÞ¼ A and ðx; y; zÞ are
parallel, we know that the surfaces have a common tangent plane at a point that is maximum for F.
Thus, rG ¼ 0isa necessary condition for a relative maximum of F at P 0 .Of course, the condition is
not sufficient. 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 find the relative maximum or minimum values of a
function Fðx 1 ; x 2 ; x 3 ; ... ; x n Þ subject to the constraint conditions ðx 1 ; ... ; x n Þ¼ 0; 2 ðx 1 ; ... ; x n Þ¼
0; ... ; k ðx 1 ; ... ; x n Þ¼ 0, we form the auxiliary function