200 MAXIMA AND MINIMA VALUES 



, du _ 



~ ~ 



Hence ~ is positive and cannot vanish for any value of x 



lying between any assigned positive value and positive in- 

 finity. We conclude that u continually increases as x changes 

 from zero to positive infinity. 



216. Maxima and minima values of an implicit function. 



Let < (x, y) =0 be an equation connecting a; and y ; it is 

 required to find the maxima or minima values of y. From 

 the given equation we know that y must be some function 

 of x, and if the equation admits of solution we can express 

 y explicitly in terms of x, and then find the maxima or minima 

 values of y by the foregoing Articles. 



But instead of solving the given equation we may proceed 

 thus : by Art. 177, 



fdu\ 

 dy _ 



dx /du\ ' 

 \dy) 



where u stands for $ (x, y). But the values of x which make 

 y a maximum or minimum must, by Art. 211, be found by 



solving the equation -f- = 0. Hence 



and this equation, combined with u = 0, will determine the 

 values of x, which may make y a maximum or minimum. 

 To determine whether such a value of x does make y a 

 maximum or minimum, we must, by Art. 211, examine the 



value of -3 . By Art. 180, since [ -p) = 0, we have 



CLX \CLxJ 



