Here's a problem that I've been stuck with for a while.
The Problem : Let $J$ be an open interval. Let $f,g:J \to \mathbb{R}$ be differentiable. Assume that $f(a)=0=f(b)$ for some $a,b$ in $J$ with $a<b$. To show that $\exists c \in (a,b)$ such that $$f'(c)+f(c)g'(c)=0$$
The only thing that I can see coming out of the given information is that $\exists y \in (a,b)$ such that $f'(y)=0$ by Rolle's theorem. I don't see how I can use the fact that both of $f(a)$ and $f(b)$ are equal to $0$ $($and not anything else$)$. We do not know anything about $g$ apart from differentiability.
Can we define a function $h$ involving $f$ and $g$ so that we may apply Rolle's theorem or Mean value theorem to get to the desired result?
Also I do suspect if we're missing any additional information here $($I tried, unsuccessfully, to come up with a counter example$)$.
Any help would be much appreciated.
Let $$ h(x)=\mathrm{e}^{g(x)}f(x). $$ Clearly, $h$ is continuous on $[a,b]$ and differentiable on $(a,b)$ and $$ h(a)=h(b)=0. $$ Rolle's Theorem provides that, there exists a $c\in(a,b)$, such that $h'(c)=0$ or $$ 0=h'(c)=\mathrm{e}^{g(c)}g'(c)f(c)+\mathrm{e}^{g(c)}f'(c)= \mathrm{e}^{g(c)}\big(g'(c)f(c)+f'(c)\big) $$ and thus $$ g'(c)f(c)+f'(c)=0. $$