MathDB

i) Let

a,b

be real numbers such that

b\leq 0

and

1+ax+bx^{2} \geq 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} -\frac{1}{a} &\text{if}\; a<0,\\ \infty & \text{if}\; a \geq 0. \end{cases}

ii) Let

f:[0,1]\rightarrow[0,\infty)

be a function with a continuous second derivative and let

f''(x)\leq0

for every

x\in [0,1]

. Suppose that

L=\lim_{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_{x\in [0,1]}|f'(x)|=L^{-1}

Problem Statement