grandes-ecoles 2010 QII.A.2

grandes-ecoles · France · centrale-maths1__pc Differentiating Transcendental Functions Higher-order or nth derivative computation

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.
Show that for all $n \in \mathbb{N}^*$, $$\sup_{x \in [-1,1]} \left| T_n'(x) \right| = 2^{1-n} n^2$$
Is this supremum attained? If so, specify for which values of $x$.

For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$.

Show that for all $n \in \mathbb{N}^*$,
$$\sup_{x \in [-1,1]} \left| T_n'(x) \right| = 2^{1-n} n^2$$

Is this supremum attained? If so, specify for which values of $x$.

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