MathDB

5

Part of 1996 IMC

Problems(1)

IMC 1996 Problem 5

Source: IMC 1996

3/5/2021
i) Let a,ba,b be real numbers such that b≤0b\leq 0 and 1+ax+bx2≥01+ax+bx^{2} \geq 0 for every x∈[0,1]x\in [0,1]. Prove that lim⁡n→∞n∫01(1+ax+bx2)ndx={−1aif  a<0,∞if  a≥0.\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases} -\frac{1}{a} &\text{if}\; a<0,\\ \infty & \text{if}\; a \geq 0. \end{cases} ii) Let f:[0,1]→[0,∞)f:[0,1]\rightarrow[0,\infty) be a function with a continuous second derivative and let f′′(x)≤0f''(x)\leq0 for every x∈[0,1]x\in [0,1]. Suppose that L=lim⁡n→∞n∫01(f(x))ndxL=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx exists and 0<L<∞0<L<\infty. Prove that f′f' has a constant sign and min⁡x∈[0,1]∣f′(x)∣=L−1\min_{x\in [0,1]}|f'(x)|=L^{-1}.
functioncalculusderivativereal analysis