MathDB

Problem 2

Part of 2008 VJIMC

Problems(2)

recurrence relation, f(f(f(x)))+4f(f(x))+f(x)=6x

Source: VJIMC 2008 1.2

6/16/2021
Find all functions f:(0,∞)→(0,∞)f:(0,\infty)\to(0,\infty) such that f(f(f(x)))+4f(f(x))+f(x)=6x.f(f(f(x)))+4f(f(x))+f(x)=6x.
fefunctional equationSequencesrecurrence relationalgebra
integral inequality involving f'(x)^2 and f(x)^(-2)

Source: VJIMC 2008 2.2

6/16/2021
Find all continuously differentiable functions f:[0,1]→(0,∞)f:[0,1]\to(0,\infty) such that f(1)f(0)=e\frac{f(1)}{f(0)}=e and ∫01dxf(x)2+∫01f′(x)2dx≤2.\int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2.
calculusintegrationinequalities