Polynomial and roots

[Papapetros Vaggelis]

If \(\displaystyle{P(z)=a_0+a_1\,z+...+a_n\,z^n\in\mathbb{C}[z]\,,a_{n}\neq 0}\) with roots

\(\displaystyle{\left\{z_1,...,z_n\right\}}\), then prove that :

\(\displaystyle{\forall\,z\in\mathbb{C}, \left(P^\prime(z)=0\implies z\in\rm{conv}\left(\left\{z_1,...,z_n\right\}\right)\right)}\).