Let $f$ and $g$ be functions such that $f''$ and $g''$ exist. Let, for some $a$c \in (a,b)$ such that
$$\frac{f(c)}{g(c)}=\frac{f''(c)}{g''(c)}$$
Let $f$ and $g$ be functions such that $f''$ and $g''$ exist. Let, for some $a$c \in (a,b)$ such that
$$\frac{f(c)}{g(c)}=\frac{f''(c)}{g''(c)}$$
Let $$h(t)=f(t)g'(t)-g(t)f'(t),$$ and apply Rolle's theorem.
(This assumes $g\ne0$ on $(a,b)$ as well.)