grandes-ecoles 2022 Q11

grandes-ecoles · France · centrale-maths2__pc Roots of polynomials Proof of polynomial identity or inequality involving roots

Let $n \in \mathbb{N}^*$ and $W$ be a monic polynomial of degree $n$. The objective of this subsection is to show that $$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }.$$ Show that $\sup _ { x \in [ - 1,1 ] } \left| T _ { n } ( x ) \right| = 1$. Deduce a monic polynomial of degree $n$ achieving equality in the above inequality.

Let $n \in \mathbb{N}^*$ and $W$ be a monic polynomial of degree $n$. The objective of this subsection is to show that
$$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }.$$
Show that $\sup _ { x \in [ - 1,1 ] } \left| T _ { n } ( x ) \right| = 1$. Deduce a monic polynomial of degree $n$ achieving equality in the above inequality.

Paper Questions

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