OF A FUNCTION OF ONE VARIABLE. 195 



If </>" (a) vanish as well as <f>(a) then, by Art. 92, 



* (a + h) = <#> (a) + </>"" (a) + f '" (a + ffl). 

 l_ l_ 



By reasoning similar to that used before, we may shew 

 that unless <f>" (a) also vanish <f) (a) can be neither a maximum 

 nor minimum value of < (x) ; but that if <"' (a) vanish and 

 <f>"" (a) be positive <f> (a) is a minimum value, and if <f>" (a) 

 vanish and <"" (a) be negative < (a) is a maximum value. 



Since this process may be continued until we arrive at 

 a differential coefficient which does not vanish when x = a, 

 we have the following result. In order that <f> (#) may have 

 a maximum or minimum value when x = a, it is necessary 

 that this value of x should make an odd number of the suc- 

 cessive differential coefficients of </> (x) vanish, beginning with 

 the first ; when this condition is satisfied < (a) is a maximum 

 value if the next differential coefficient be negative and a 

 minimum value if it be positive. 



212. It is to be observed that in the above demonstration 

 we have used 6 to denote a fraction less than unity, and it 

 is not to be assumed that the same fraction is denoted when- 

 ever the symbol is used. Also we have supposed as usual 

 that none of the functions </>'(), <"(), ... are infinite. We 

 shall shew hereafter, that maxima and minima values 

 may occur when <' (x) = oo , as well as when <' (x) = : see 

 Art. 214. 



213. Suppose that when x = a, the function <j> (x) has a 

 maximum or minimum value, and that <f> n (a) is the first 

 differential coefficient that does not vanish, n being even. 

 By Art. 92, since <}>' (a), $"( a )> ... all vanish up to $ n ~*(a] 

 inclusive, we have 



where 6 and t are proper fractions. 



From these values of </>' (a + h] and </>' (a h} we see that 

 <f>' (x) changes sign as x passes through the value a. If we 

 suppose x to increase and pass through the value a, then 



02 



