Let $n$ be a non-zero natural number. Deduce that, for all $P$ in $\mathbb{C}_n[X]$, $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$
Let $n$ be a non-zero natural number. Deduce that, for all $P$ in $\mathbb{C}_n[X]$,
$$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$