Let $p(x)=(x-α_1)^{m_1}\cdots(x-α_r)^{m_r}$ be a polynomial of degree $n$, and the roots $α_1,\ldots,α_r$ are distinct.
Let $c$ be a number for which $p'(c)≠0$.Show that there exists a zero $α$ of $p(x)$ satisfying $$|α-c|≤n\left|\frac{p(c)}{p'(c)}\right|.$$
I'm working on a question related to root-finding for non-linear equations. I have an idea of using the mean value theorem for this polynomial but I don't know the exact way of doing it.