194 MAXIMA AND MINIMA VALUES 



If <f>'(x) be not zero we can give such a value to h that 

 the sign of 



shall for that value of h, and all inferior values of h, be the 



same as the sign of hfi (x), because - <" (x + 6h) can always 



z 



be made less than <f> (x) by taking h small enough. In this 

 case 



<f> (x + h) <f> (x) 



and <f> (x h) <f> (x) 



have different signs, and therefore <f> (x) has neither a maxi- 

 mum nor minimum value. 



Hence, as the first condition for the existence of a maxi- 

 mum or minimum value of < (x), we must have 



f(*0=o ........................... (i). 



Let a be a value of x deduced from equation (1), so that 



<'(a)=0. 

 We have now, by Art. 92, 



</> (a + A) = * (a) + 1 f (a) +| *'" (a + 0A). 



Suppose <"() not zero; then by giving to h some value 

 sufficiently small, the sign of 



will be the same as that of , <j>'(a), or of <$>"(a), for that 



If 

 value of h and all inferior values ; 



therefore $ (a + A) <f> (a) 



and <p(a K) (f> (a) 



have the same signs. 



If then </>" (a) be positive <j> (a) is a minimum value of 

 <(#) ; if <"(a) be negative <f> (a) is a maximum value of <j> (x). 



