238 EXAMPLES OF MAXIMA AND MINIMA. 



The first expression becomes 1, the second becomes 0, 

 and the third becomes sin 2 a, when the assigned values of 



T-J.-J.LJ TT d?u <Fu ( d l u \*> 



x and y are substituted. rLence ^-, - 7 -_ -= y- is positive, 



dx* dy* \dxdyj 



and u is a maximum. 

 3. Suppose u 



TT d u , du 



Mere -5- = 0, and -j- = 0, give as one pair of values x = 0, 



y = 0. And these values make 



cPu ffu d*u ^, 



-i-^ = 2a, , , =0, 3-2 = 25; 

 ax" dx dy dy* 



therefore u has then a minimum value. 

 Another pair of values is given by . 



aj = 0, 



and b ax z by 3 = 0, 



that is, x = 0, and y = l. 



With these values we have 



* * A d*u 1 _! 



' df = 



Hence, if a is less than, b, we have a maximum value of u, 

 and if a is greater than b, we have neither a maximum nor 

 a minimum. 



There is only one other solution, namely, that found by 

 combining 



y = 0, and a ax* by a = 0' ) 



therefore y = 0, and x= 1. 



