I know the proof of the theorem by applying the mean value theorem, but the teacher asked us to prove it by integration by parts. However, I insist that it's not possible, because integration by parts requires more conditions which are not satisfied by the hypothesis. Could help me to construct an argument or a prove if possible?
Let be $f(x):[a,b] \rightarrow \mathbb{R}$ and $g(x)$ a nonnegative integrable function on $[a,b]$, then there exists $c \in [a,b]$ such that:
$$\int_{a}^{b}f(x)g(x)dx = f(c)\int_{a}^{b}g(x)dx$$
Integration by parts that I know: Let $F,G$ be differentiable on $[a, b]$ and let $f:= F'$ and $g := G'$ belong to $R_{[a,b]}$ then,
$$\int_{a}^{b}fG = \left.FG\right|_{a}^{b} - \int_{a}^{b} Fg$$