I'm sorry for my bad english, I translated it from a French exercice.
Let $f$ be a function of class $C^3$ defined on $[a,b]$, and $c=(a+b)/2$. We want to approximate the quantity $d(f) = f''(c)$ by an expression of the following form :
$\delta(f) = \lambda_0 f(a) + \lambda_1 f(c) + \lambda_2 f(b)$
which is such as $d(Q) = \delta(Q)$ for any polynomial $Q$ of degree less than or equal to 2.
Let $P$ be the Lagrange interpolation polynomial of $f$ on the points $(a,c,b)$ and $r$ the interpolation error $r(x) = f(x) - P(x)$. Show that $d(f) - \delta(f) = r''(c)$.
Deduce that $|d(f) - \delta(f)| \leq \dfrac{b-a}{2} \sup_{x \in [a,b]} |f'''(x)|$.
(We found previously that the divided difference of the function f on the points $(a,c,b)$ verify $f[a,c,b] = \delta(f)/2$)
I'm blocked on this question. Could someone help me ? Thank you in advance.