Let $f : [a, b ] \rightarrow \Bbb{R}$ be a continuous function and $g$ Riemann integrable such that $g(x) \geq 0$ for all $x \in [ a, b] $ prove that if $\int_{a}^{b} f \cdot g = f(a) \int_{a}^{b} g$ then there exist $c \in (a, b) $ such that $f(c) = g(c) $
My teacher suggested to use the mean value theorem for integrals.
I know that the mean value theorem for integrals is only applicable to continuous functions so $\int_{a}^{b} f = (b-a) f(c)$ for some $c\in(a,b)$. My problem is that I don't see how to use the hypothesis to reach the conclusion, only that advice would suffice.
This is false. If $f$ is any constant function then the hypothesis is satisfied. If the constant value of $f$ is not in the range of $g$ then we cannot have $f(c)=g(c)$. For a specific example take $f \equiv 1$ and $g \equiv 2$.