IMC 1996 · Problem 5

(i) Let $a$, $b$ be real numbers such that $b \le 0$ and $1 + ax + bx^2 \ge 0$ for every $x$ in $[0,1]$. Prove that

$$\lim_{n \to +\infty} n \int_0^1 (1 + ax + bx^2)^n dx = \begin{cases} -\dfrac{1}{a} & \text{if } a < 0, \\[4pt] +\infty & \text{if } a \ge 0. \end{cases}$$

(ii) Let $f : [0,1] \to [0,\infty)$ be a function with a continuous second derivative and let $f''(x) \le 0$ for every $x$ in $[0,1]$. Suppose that $L = \lim\limits_{n \to \infty} n \int_0^1 (f(x))^n\,dx$ exists and $0 < L < +\infty$. Prove that $f'$ has a constant sign and $\min\limits_{x \in [0,1]} |f'(x)| = L^{-1}$.