grandes-ecoles 2010 QII.C

grandes-ecoles · France · centrale-maths1__pc Proof Direct Proof of an Inequality

Let $P \in E_n$. Show that: $$\sup_{x \in [-1,1]} \left| P'(x) \right| \leq n^2 \sup_{x \in [-1,1]} |P(x)|$$ (One may use the trigonometric polynomial $T(\theta) = P(\cos(\theta))$.)

Let $P \in E_n$. Show that:
$$\sup_{x \in [-1,1]} \left| P'(x) \right| \leq n^2 \sup_{x \in [-1,1]} |P(x)|$$
(One may use the trigonometric polynomial $T(\theta) = P(\cos(\theta))$.)

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2