Could you please help me to answer this question for freshmen students :
Let $f:I=[a,b]\rightarrow\mathbb{R}$ be a function of class $2$ and suppose that there exist $\ell\in I$ such that $f(\ell)=\ell$ and $f'(\ell)=0$. Without using Taylor-Lagrange theorem, prove that there exist $c\in]a,b[$ such that:
$$f(a)=\ell + \frac{(a-\ell)^{2}}{2}f''(c)$$
Noting that, we can use Rolle's theorem, mean value theorem, etc..