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  3. 设f(x)在[0,1]上连续且单调减少，且f(x)＞0，证明：.

设f(x)在[0,1]上连续且单调减少，且f(x)＞0，证明：.

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问题 设f(x)在[0,1]上连续且单调减少，且f(x)＞0，证明：.
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答案[*]等价于∫01f2(x)dx∫01xf(x)dx≥∫01f(x)dx∫01xf2(x)dx 等价于∫01f2(x)dx∫01yf(x)f(y)[f(x)－f(y)]dy≥0 令I=∫01dx∫01yf(x)f(y)[f(x)－f(y)]dy 根据对称性,I=∫01dx∫01xf(x)f(y)[f(y)－f(x)]dy 2I=∫01dx∫01f(x)f(y)(y－x)[f(x)－f(y)]dy 因为f(x)＞0且单调减少，所以(y－x)[f(x)－f(y)]≥0,于是2I≥0，或I≥0 所以 [*]
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