MathDB

15

Part of 2006 China Second Round Olympiad

Problems(1)

2006 China Second Round Olympiad Test 1 #15

Source:

9/28/2014
Suppose f(x)=x2+af(x)=x^2+a. Define f1(x)=f(x)f^1(x)=f(x), fn(x)=f(fn−1(x))f^n(x)=f(f^{n-1}(x)), n=2,3,⋯n=2, 3, \cdots, and let M={a∈R∣∣fn(0)∣≤2,for any n∈N}M=\{a\in\mathbb{R}| |f^n(0)|\le 2, \text{for any } n\in\mathbb{N}\} . Prove that M=[−2,14]M=[-2, \frac{1}{4}].