Originally Posted by

**matt.qmar** Hello!

I am working on this problem:

Suppose $\displaystyle f$ and $\displaystyle g$ are continuous and positive on some interval $\displaystyle [a,b]$.

I am trying to show that $\displaystyle \exists\ \zeta \in [a,b]$ such that $\displaystyle \int_a^bf(x)g(x)dx = f(\zeta)\int_a^bg(x)dx $

All I have so far is that since both $\displaystyle f$ and $\displaystyle g$ are continuous, then their product is continuous (and therefore Riemann-integrable) but I'm not sure how to select this zeta. Whole thing looks like some sort of mean value theorem for products of functions?!

Any help appreciated. Thanks!!