Exercise 4, § 2

a) Prove the relation

(use the fact that $b_{2n-1} = 0$ for $n > 1$, and the relation )

b) Show that

(use exerc. 3).

c) Show that, for $n$ even, $B_n(X)$ has two roots in the interval $[0, 1]$ of $\mathbf{R}$, and that for $n$ odd $> 1$, $B_n(X)$ has a simple root at the points $0, \frac{1}{2}$ and $1$ and does not vanish at any other point of $[0, 1]$ (use $b$) and the relation $B'_n = n \tilde{B}_{n-1}$).

d) Deduce from c) that, for $n$ even, the maximum of $|B_n(x)|$ on the interval $[0, 1]$ is $|b_n|$, and that for $n$ odd, if $a_n$ is the maximum of $|B_n(x)|$ on $[0, 1]$, then

(use the mean value theorem).