Elementary question about roots with multiplicity of a polynomial.

Question

How do i prove following lemma in general:

Let $x_0$ be a root with multiplicity $m$ of a polynomial $f$, then $\forall k\in \{0,1,...,m-1\}: f^{(k)}(x_0)=0$.

Answer 1

HINT:

$$f(x) = (x-x_0)^mg(x)$$

where $g(x_0) \not = 0$.