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 $1 \leqslant k \leqslant n$. Now prove that the rational function

is increasing in 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)), f is convex for x < c₁ and concave for x > c₁. Deduce that when a ranges over (−1, 1), the length of the greatest interval containing the zeros of the kth derivative of (x − a)P(x) attains its greatest value when a = 1 or a = −1.