EXAMPLES OF MAXIMA AND MINIMA. 259 



Therefore, from (7), (8), and (9), 



\l 



<Y 2 . 



**/ ~~ tj i7 * 



U (T 



and thus, from equation (3), 



3 + 2 + 'J 1 2 = 0. 



This equation is a quadratic in u?, from which two values 

 of w 2 can be determined, one of which will be a maximum 

 and the other a minimum. It is obvious that a maximum 

 and a minimum value of w a must exist, for sc, y, z, cannot all 

 vanish simultaneously, and no one of them can be greater than 

 unity ; hence w a must lie between the limits and a 2 + 6 2 + c 2 . 



4. Find the values of x, y, z, when x*yz z is a maximum 

 or minimum, subject to the condition 



We have, putting u for 



to?yz*Dx + x'JDy + Zx'yzDz = 0, 



(iDx Dy , 

 u\ 1 -- H r "P.O. 



I syt ,, i 01 1 



( x y z ) 



tfxDx + $by*Dy + 2z*J)z = 0. 



4 



- + \a*x = 0, 

 x 



1 



- + 3X&J/ 0, 



y 



i 8 _ 



z 



S2 



