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# Use the weighted mean value theorem to establish an inequality

Use the weighted mean value theorem (Theorem 3.16 in Apostol) to prove: Recall the weighted mean value theorem:

For functions and continuous on , if never changes sign in then there exists such that Proof. Let Then substituting our definitions of and , Since and are continuous and does not change sign on we may apply Theorem 3.16, Since is strictly decreasing on , we have Then, 1. Mihajlo says:

It seems safer to do integration on the product of g and f, as in the proof of 3.16 (and not like done here).

• Mihajlo says:

Sorry, the way you proved it seems valid (apart from the mentioned error in the last row) and probably the intended way, because it uses the required theorem.

2. Jesse B says:

The inequality you end up with is not the one the question is asking for; there is an extra factor of f(c).

• jon says:

remember that f(c)(integral of g(x)) = (integral of f(x)g(x)) which is the original equation. In other words both the function he ended up with and the one being asked for are equivalent.

*edit although he did make a typo and rewrote f(c) in the final line when he had already factored it back in.