Exercise 24, § 4

Let $P(x)$ be a polynomial of degree $n$ with real coefficients all of whose roots are real and contained in the interval $[-1, 1]$. Let $k$ be an integer such that $1 \leq k \leq n$. Show that the rational function

is increasing on every interval of $\mathbf{R}$ on which it is defined: if $c_1 < c_2 < \cdots < c_r$ are its poles (contained in $[-1, 1]$), then $f$ is convex for $x < c_1$ and concave for $x > c_r$. Deduce that when $a$ runs through $[-1, 1]$ the length of the largest interval containing the zeros of the $k^{th}$ derivative of $(x-a)P(x)$ attains its largest value when $a = 1$ or $a = -1$.