积分证明题设函数f(x)∈C［0,1］∩D（0,1）,且f(0)=0,0＜f'(x)＜1,证明［∫(0,1)f(x)dx］^2>∫(0,1)f^3(x)dx.后面那个是三阶导数

Question

积分证明题
设函数f(x)∈C［0,1］∩D（0,1）,且f(0)=0,0＜f'(x)＜1,证明［∫(0,1)f(x)dx］^2>∫(0,1)f^3(x)dx. 后面那个是三阶导数

Answer 1

［∫(0,1)f(x)dx］^2>∫(0,1)f^2(x)dx的证明亲看下面的链接http://zhidao.baidu.com/question/506265595.html?oldq=1&from=evaluateTo#answer-1272614564
证明∫(0,1)f^2(x)dx>∫(0,1)f^3(x)dx
构造L(t)=∫(0,t)f^2(x)dx-∫(0,t)f^3(x)dx
L(0)=0
L'(t)=f^2(t)(1-f(t))
由于f'(x)＜1
∫(0,t)f'(x)dx