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,

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

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.