202 MAXIMA AND MINIMA VALUES 



its real value. Forming the derived equations from the 

 given equation, we have 



When we put x = 0, y = 0, in these, the first equation gives 



2? = 0, and the second equation gives -j^ = . Hence, 

 dx dx* 3a 



when x= 0, and y = Q, we have y a minimum. 



217. If the values of x and y found from w = and 



( -3^ ) = 0. make \ vanish, then in order that they may 

 \dxj dx 



make y a maximum or minimum, it will be necessary that 



d z y 



-ri should also vanish. This can be tested by making use 



73 

 of the value of 3-5 given by Art. 184; and by obtaining 



CMW 



a formula for -~ similar to that for -.-% iust referred to, we 

 dx 4 dx 3j 



d*y 

 can ascertain whether -^ is positive or negative for the 



dX 



specific values of x and y. On account however of the 

 complexity of the general formulae for -^ and -rj[ , it is 



CuOC CttC 



preferable to determine them in any example directly by the 

 method of Art. 184, rather than to quote the results of that 

 Article. 



218. Suppose u = (f)(x, y} and -fy (x, y) = ; so that y is 

 a function of x by the second equation, and therefore from 

 the first equation u is a function of x ; required the maxima 

 and minima values of u. We may proceed theoretically thus : 

 by solving the equation ^r (x, y] = 0, obtain y as a function 

 of a;; substitute this value of y in < (x, y] ; then u becomes a 

 function of x only, and its maxima and minima values can 

 be found by previous rules. But we may avoid the difficulty 

 of solving the equation i/r (x, y) = 0, thus. 



