>>18

>>48のとちょっと違う方法で解けてみます。



0 = ∫［0,1] x^2・f(ｘ）ｄｘ
=[ｔ・∫[0,t] s・f(s）ｄs][t=0,1] - (∫[0,1]∫[0,t] s・f(s）ｄs dt ) (部分積分法)
=(∫[0,1] s・f(s）ｄs - 0) - ∫[0,1]∫[0,t] s・f(s）ｄs dt
= 0 -∫[0,1]∫[0,t] s・f(s）ｄs dt (∵∫[0,1] s・f(s）ｄs = 0)
= -∫[0,1]∫[0,t] s・f(s）ｄs dt

∴∫[0,1]∫[0,t] s・f(s）ｄs dt = 0

∃ａ∈[0,1] s.t. ∫[0,ａ] s・f(s）ｄs = 0