

OF AN IMPLICIT FUNCTION. 201 



Hence we have this rule : To find the maxima or minima 

 values of y, which is an implicit function of x determined by 

 u = 0, we must find values of x and y which satisfy u = 



and (-,-) = 0. If when these values are substituted in 



\dxj rdu 



(dyj 



the fraction is positive, we have obtained a maximum value 

 of y ; if the fraction be negative, we have a minimum value 

 of y. 



Example. If x a - Zaxy + y 3 = (1), 



find the maxima or minima values of y. 



Here -f- = -if- ; 



ax y ax 



therefore ay x* = Q (2). 



Combining (1) with (2), we have 



therefore x = 0, 



or x = a /2. 



The corresponding values of y are 



. 



/d*u\ 



\ rl 2 / 



If we substitute the values x = a f/2. v = a 54. in ^ , 



(LLit \ 

 Ty) 



that is, in -^ - -, we obtain . Hence there is a 

 3 (y* ax) a 



maximum value of y. The values x = 0, y = 0, which make 



the numerator of -^ vanish, also make its denominator vanish : 



dv . 



thus -j - assumes an indeterminate form, and we must discover 



