International Mathematics Competition
for University Students
2020

IMC2020: Day 1, Problem 5

Problem 5. Find all twice continuously differentiable functions \(\displaystyle f:\mathbb{R}\to(0,+\infty)\) satisfying

\(\displaystyle f''(x)f(x)\geq {2(f'(x))^2} \)

for all \(\displaystyle x\in\mathbb{R}\).

Karen Keryan, Yerevan State University & American University of Armenia, Yerevan

Hint: The expression \(\displaystyle f''\cdot f - 2(f')^2\) is a part of the second derivative of some fraction.

  Solution