Let $f(x)$ be twice continuously differentiable on $[a, b]$ and $f(a)=f(b)=0$. Show that there exists $c\in[a, b]$ such that $$\int_a^b f(x)dx=-\frac{f''(c)}{12}(b-a)^3 $$
I have tried using the Taylor expansion of $f(x)$ at $x=a$ and $x=x_0$ where $f'(x_0)=0$ (Rolle's theorem), but nothing seems to work. Any hint will be appreciated.
