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### Polynomial and roots

Posted: Mon Apr 18, 2016 8:52 pm
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 :

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