^Q < Art. 4. — M. Kuniyetla : 



(i) The case in which x<f><xo', or p = Ax(\-(>), where 

 ^>0, ^>0, /><1. 



In this case, by H-lemma 33 and Lemma 3, the function 



(39) <f = '^^^ 



x{).^a'{x)] 



is a steadily decreasing function of .r, except for the value x = 6, 

 throughout the interval 0<x<ç; and we divide the integral (37) 

 into the three parts 



(40) /,(/) = /""V / '' + / ' = J,'''+J,'''+ J:'''. 



.1 ./ 6-1 ' e^c 



(ii) The case in which x'<i><x, oy p = Ax{\.^'(>\ where 

 yl > 0, <, > 0, i> < 1. 



In this case, by Lemma 3, the function <f has one stationary 

 value, which is a minimum given Ijy x = a, a being the root of 



the equation ^ — ^- Herel)y « is greater than Ö, so that the 

 function ^ is a steadily decreasing function of x in the interval 

 < a; < a, except onl}^ for the value x = d, and it is a steadily 

 increasing function of x in the interval a<x<:.ç. As it will be 

 proved presently, we have 



Hence we divide the integral -/, into the four parts 



(41) /,(/)-/■'' + /"' + r + /'"^ = //^) +//•-') +j/^)+j/^>. 



./ ./ 9-: ,/ e c J a 



23- Integrals J/'^ and J!'''\ As x increases from to d-z, the 

 function y—Xx + o{x) decreases from x {o k{d — t) + (T{d — t). which 

 is large and positive when is small and £ smaller. Also 



//'> = /"' r - , . '"^^T-l «*« '/ ^ '/• 



' .//(»-c)f»(fl-c; L x[Ä + a'{;x)} A ■' ■' 



The factor— cJ.L o which multiplies cosy is i)ositive and 

 x{k + o'(x)] ' -' ' 



monotonie, as we have already seen. Hence, by the Second Mean 

 A'alue Theorem, we obtain 



