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Let $f$ be a nonnegative continuous function on $[0,1]$ , and $f$ nondecrease. Then for any $0<\alpha<\beta<1$ , we have $\int_{0}^{1}f(x)dx\geq\frac{1-\alpha}{\beta-\alpha}\int_{\alpha}^{\beta}f(x)dx$ , and $\frac{1-\alpha}{\beta-\alpha}$ is the biggest number satisfying the inequality.

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Please formulate your question as a question. What do you want to prove? Also sketch what you have tried so far. –  Julian Kuelshammer Jan 11 '13 at 15:45
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1 Answer 1

Hint:

Show that the average value of $f(x)$ on $[\alpha,1]$ is at least the average value of $f(x)$ on $[\alpha,\beta]$.

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I want to know how to prove that $\frac{1-\alpha}{\beta-\alpha}$ be the biggest number satisfying the inequality –  user56927 Jan 11 '13 at 16:00
    
I know. I just don't want to do your whole homework problem for you. –  Zarrax Jan 11 '13 at 16:02
    
I can pove the inequality , I only can't konw how to explain the last one. –  user56927 Jan 11 '13 at 16:06
    
Equality holds for the function that is 1 on $[\alpha,1]$ and zero on $[0,\alpha)$. Approximate this function by continuous functions. –  Zarrax Jan 11 '13 at 16:23
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