Exercise 8, § 3

Let $f$ be a real function defined and $n$ times differentiable on an interval $I$, let $x_1, x_2, \ldots, x_p$ be distinct points of $I$, and $n_i$ ($1 \leq i \leq p$) be $p$ integers $> 0$ such that

Suppose that at the point $x_i$ the function $f$ vanishes together with its first $n_i - 1$ derivatives for $1 \leq i \leq p$: show that there is a point $\xi$ interior to the smallest interval that contains the $x_i$ and such that $f^{(n-1)}(\xi) = 0$.