OF AN IMPLICIT FUNCTION. 203 



By Art. 172, we have 

 du 

 dx 



_ fdu\ /du\ dy 

 ~ \dx) \dy] dx ' 



Also, putting v for ty (x, y), we have, by Art. 177, 



dy _ \dx) m 

 dx fdv\ ' 



fdu 



c du fdu\ \dy) \dx 



therefore ^ ' > v ^ 



ax 



du _ fdu\ 

 dx ~ \dx) 



\dy 



Hence, the values of x and y that render u a maximum 

 or minimum must be sought among those that satisfy simul- 

 taneously 



fdu\ fdv\ _ fdu\ fdv\ _ 

 \dx) \dy) \dy) \dx) 



and ty (x, y) or v = 0. 



The value of -= ,- must then be found by Art. 176, and 



we must examine whether the specific values of # and y 

 render this positive or negative, in order to determine whether 

 u is a minimum or a maximum. 



Example. u = x* + y*, 



while (x - a) 2 + (y - J) 2 - c 2 = 0, or v = 0. 



du 

 -y- 

 dy 



dv 



fdu\ 



Here ( -7- = 2a?, 

 \dx) 



Hence x(y l}y(x 



therefore ay = bx. 



