38 



Art. 4. — M. Kxiniveda: 



(i) Let o<x or r = Ax{V-v^p), where A>0, p^O, ^<1. Then 

 <p is a steadily increasing function of x throughout the interval 

 0<x<ç. Hence, by the Second Mean Value Theorem, 



Therefore «/.(z) = 0(1//). 



(ii) Let a:</><a:(T' or /> = ^Ja'(l-^o), where ^>0, /^>0, ô<]. 

 Then </ has one stationary value in the interval < ic < ?, which is 

 a maximum given by x = a, a l>eing the root of the equation 



dx 



\\q now write 



•^=« = (/-I + ./■' ) .{ÄW] "^ ' "■' = ^='^^=" 



say, where /9 = la—a{ri). Tlien, by another application of the 

 Second Mean Value Theorem, 





Therefore J-! = ^^p^^^'^^^^^j • (1) = [^.(a)} . 



Similarly we obtain 



Hence we have 



The same argument applies to the integral J^i^^)- Hence, 

 if p<x or o = .-la-(l + /')j whei'o .4>0, ,tj^0, /^ < 1, 



