Exercise 4, § 2

a) Prove the relation

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

b) Prove that one has

(use Exercise 3).

c) Prove 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 $B'_n = n B_{n-1}$).

d) Deduce from c) that, for $n$ even, the maximum of $|B_n(x)|$ in the interval $(0,1)$ is equal to $|b_n|$, and that for $n$ odd, if $a_n$ is the maximum of $|B_n(x)|$ in $(0,1)$, one has

(use the theorem of finite increments).