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- May 23rd 2008, 05:16 AMszpengchaoMean Value Theorem for Integrals
- May 23rd 2008, 08:32 AMThePerfectHacker
Since $\displaystyle f$ is continous it attains its maximum and minimum.

This means there are $\displaystyle k_1,k_2$ so that $\displaystyle f(k_1)\leq f(x)\leq f(k_2)$.

Since $\displaystyle g(x)\geq 0$ multiplication does not reverse inequality, $\displaystyle f(k_1)g(x)\leq f(x)g(x)\leq f(k_2)g(x)$.

Now integrate both sides, this is possible since $\displaystyle g$ is continous.

And finally the last statement follows by indetermediate value theorem for continous functions.