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heyinprc(feixiaolin´ú·¢): ½ð±Ò+10 2013-12-10 07:50:41
heyinprc: ½ð±Ò+5, ¡ï¡ï¡ïºÜÓаïÖú, 1£©Óë3£©ÎÒȷʵÊÇÕâÑùдµÄ£¬²»¹ý1£©ÎÒ¾õµÃ×Ô¼ºÕâÑùдºÃÏñ²»Ì«ÑϽ÷¡£ 2013-12-10 21:01:22
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heyinprc(feixiaolin´ú·¢): ½ð±Ò+5 2013-12-11 20:28:29
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4Â¥: Originally posted by heyinprc at 2013-12-10 21:02:19
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ÀûÓà Sum_{n=1}^Infinity n* x^n = x/ (1-x)^2, Ö¤Ã÷
f(0)= e/ (e-1)^2 > 2/e;  f(Pi) = - e/ (e+1)^2. ½øÒ»²½Ö¤Ã÷
´æÔÚ0<x<Pi ʹµÃ f¡¯(x)= ( f(0)-f(Pi) ) /Pi = 2e/Pi * (e^2+1)/(e^2-1) > 2e/Pi.  

µÚÒ»Ì⣺ÎÒÃǼÙÉèÃüÌâ²»³ÉÁ¢£¬ ²»·ÁÉèf(x) + f¡¯¡¯(x) >0. (·ñÔò¿ÉÒÔ¿¼ÂÇh(x)= -f(x), ËüÂú×ãͬÑù¼ÙÉè, µ«ÊÇ h(x)+ h¡¯¡¯(x) >0 ). ²»·ÁÉè f¡¯(0) >0, ·ñÔò¿ÉÒÔ¿¼ÂÇ f(-x).

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Õâ¸öº¯ÊýÊÇÌØÒâÉè¼ÆÂú×ã g(0)=g¡¯(0)=0, g¡¯¡¯(x)= f(x) +f¡±(x).
g¡¯¡¯(x) >0 Ö¸ g(x) ÊÇ͹º¯Êý£¬ ÔÚRÉÏÓÐΨһ×îСֵ g(0)=0.

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Integral_0^ a (a -t)dt  > f(0) -f(a) +  a^2.  ¼´ 1/2*a^2 > f(0) -f(a) + a^2.
Èç¹û f(0) >=0, , f(a) > f(0)/2 + ( f(0) + a^2)/2 >=2, ²»¿ÉÄܵġ£
ËùÒÔ f(0) <0.
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Õâʱ¿¼ÂÇ Integral_d^ x [(x-t)*f(t)] dt + f(x)- f(d) -x * f¡¯(d). ²¢ÇÒÈ¡ x=f¡¯(d),
ÄÇôӦ¸ÃÓÐ f ( f¡¯(d)) > 1/2 * f¡¯(d)^2. ÕâÊDz»¿ÉÄܵġ£

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heyinprc

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