MathDB

Problem 1

Part of 2012 VJIMC

Problems(2)

|f'(x)|≠1, f has a fixed point and f(b)=1-b

Source: VJIMC 2012 1.1

5/31/2021
Let f:[0,1]→[0,1]f:[0,1]\to[0,1] be a differentiable function such that ∣f′(x)∣≠1|f'(x)|\ne1 for all x∈[0,1]x\in[0,1]. Prove that there exist unique α,β∈[0,1]\alpha,\beta\in[0,1] such that f(α)=αf(\alpha)=\alpha and f(β)=1−βf(\beta)=1-\beta.
functioncalculus
given limsup, integral converges

Source: VJIMC 2012 2.1

5/31/2021
Let f:[1,∞)→(0,∞)f:[1,\infty)\to(0,\infty) be a non-increasing function such that lim sup⁡n→∞f(2n+1)f(2n)<12.\limsup_{n\to\infty}\frac{f(2^{n+1})}{f(2^n)}<\frac12.Prove that ∫1∞f(x)dx<∞.\int^\infty_1f(x)\text dx<\infty.
real analysiscalculusintegration