Let $f,g:[a,b]\to\mathbb{R}$ both increasing, twice differentiable and $g$ is also convex, $f(a)=0$, $g(b)=0$, $f^\prime(a)=0$. Prove that there exists $c\in(a,b)$ such that $$f''(c)g(c)+f(c)g''(c)+f'(c)g'(c)=0$$
I found that if we denote by $h(x)=f''(x)g(x)+f(x)g''(x)+f'(x)g'(x)$, we have that $h$ has the intermediate value property and, also, $h(a)\leq 0$ and $h(b)>0$. I'm stuck only if $h(a)=0$.