Let $p_n \in P_n[x]$ be the unique polynomial interpolating $f$ at $x_0, \dots, x_n$. Define $e(x) := f(x) - p_n(x)$. Then $e \in C^n[a, b]$ and $e$ vanishes at the $n + 1$ distinct points $x_0, \dots, x_n$. By repeated application of Rolle's theorem: $e$ has $n + 1$ zeros, so $e'$ has at least $n$ zeros in $(a, b)$; then $e''$ has at least $n - 1$ zeros; continuing, $e^{(n)}$ has at least one zero $\xi \in (a, b)$. At this point: \begin{align*} 0 = e^{(n)}(\xi) = f^{(n)}(\xi) - p_n^{(n)}(\xi). \end{align*} Since $p_n$ has degree $n$ with leading coefficient $f[x_0, \dots, x_n]$, its $n$-th [derivative](/page/Derivative) is the constant $n!\, f[x_0, \dots, x_n]$. Hence $f^{(n)}(\xi) = n!\, f[x_0, \dots, x_n]$, giving the result. $\blacksquare$