EXAMPLES OF MAXIMA AND MINIMA. 239 



Here we should find that if a is less than b, there is 

 neither a maximum nor a minimum, and if a is greater than 

 b, there is a maximum value of u. 



If in this example a = b, we arrive at the anomalous case 

 considered in Art. 235. 



4. Let u = smx + siny + cos (x + y), 



du . . N 



-j- = cos as sin (x + y), 



du / \ 



- = cos v sin (x + ?/). 

 ay 



If -j- and jr vanish, we must have 

 o# at/ 



cos x = cos y = sin (as + y}. 



These equations admit of numerous solutions. For ex- 

 ample, 



if cos x cos y, 



we have x = y, as one solution. 



Hence we have cos x = sin 2# 



= 2 sin x cos x ; 

 therefore, either cos x = 0, or sin x = \. 



7T ; 



If we take the first, and put x = y=-, we have neither 

 a maximum nor a minimum ; if we put 



_37T 



we obtain a minimum. 



If we take sin x = ^, and put 



we obtain a maximum value for u. 



