Exercise 9, § 3

With the same notation as in exerc. 8 suppose that $f$ is $n$ times differentiable on I but otherwise arbitrary. Let $g$ be the polynomial of degree $n - 1$ (with real coefficients) such that at the point $x_i$ ($1 \leq i \leq p$) both $g$ and its first $n_i - 1$ derivatives are respectively equal to $f$ and its first $n_i - 1$ derivatives. Show that we have

where $\xi$ is interior to the smallest interval containing the points $x_i$ ($1 \leq i \leq p$) and $x$. (Apply exerc. 8 to the function of $t$) where $a$ is a suitably chosen constant.)