84 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and when h is sufficiently small, these will be of opposite 

 signs if n be odd, and there will be neither maximum nor 

 minimum but of the same sign if n be even, and that 

 sign the same as the sign of f n (a), which must therefore 

 be negative for a maximum and positive for a minimum. 



Hence, for a maximum or minimum value f (a) of /"(#), 

 an odd number of the derived functions /' (a), /"(), 

 /'"(a), ... must vanish, and if so, the sign of the first, 

 which remains finite, will determine whether f(a) is a 

 maximum or minimum, viz. if it be negative, f(ci) is a 

 maximum, if positive, /(a) is a minimum. 



As a simple example of the use of this method, suppose 

 y is a function of x given by the equation x s + y 3 -$axy = 0, 



hence (y* - ax) -^ + x* - ay = 0, and -~- o.v j' {x) = when 

 x 2 = ay, and therefore 



y 3 3axy x 3 = Saxy axy = 2axy, 

 therefore y Q or y* = 2ax. If we take # = 0, we have 



x = 0, and therefore -&- becomes - (one of the true values is 

 dx 



and gives a minimum), but taking y i = 2aaj, therefore 

 y* = 4aV* = 4 3 ?/, or y = 4a 3 , and therefore x 3 = 2d j . To 



cL*ii 



find the corresponding value of/" (a?) or -,-;, differentiate 



the equation again, rejecting the terms involving ~ as a 



dx 



factor since it is 0, and we have (y* ax) y-f + 2x = 0, 







or since y 2 = 2ax, ax 2 +2x = 0, or -y- = -- ; therefore 



the value 4^a is a maximum value of y. (To find whether 

 the value y is a maximum or minimum, it is better to 

 consider the equation directly ^ when y is small the term y 3 

 may be neglected compared with the others, and we have 

 approximately x 3 3axy = 0, or a; 2 - Say = as an approxi- 

 mation form of the relation between x and y. Here we 



