TITLE: Integral of product of two functions in terms of the integral of the other function
QUESTION [9 upvotes]: Problem:

Let $f$ and $g$ be two continuous functions on $[ a,b ]$ and assume $g$ is positive. Prove that $$\int_{a}^{b}f(x)g(x)dx=f(\xi )\int_{a}^{b}g(x)dx$$ for some $\xi$ in $[ a,b ]$.

Here is my solution:
Since $f(x)$ and $g(x)$ are continuous, then $f(x) g(x)$ is continuous. Using the Mean value theorem, there exists a $\xi$ in $[ a,b ]$ such that  $\int_{a}^{b}f(x)g(x)dx= f(\xi)g(\xi) (b-a) $ and using the Mean value theorem again, we can get $g(\xi) (b-a)=\int_{a}^{b}g(x)dx$ which yields the required equality. 
Is my proof correct? If not, please let me know how to correct it.

REPLY [4 votes]: The integrals on both sides of the problem are well defined since $f$ and $g$ are continuous, and $g$ is positive so $ \displaystyle \int^b_a g(x) dx > 0.$ Thus there exists some constant $K$ such that $$ \int^b_a f(x) g(x) dx = K\int^b_a g(x) dx . $$
If $\displaystyle  K > \max_{x\in [a,b]} f(x) $ then the left side is smaller than the right. 
If $\displaystyle  K < \min_{x\in [a,b]} f(x) $ then the left side is larger than the right.
Thus $ K \in f( [a,b] ).$