International Mathematics Competition
for University Students
2019

IMC2019: Day 2, Problem 6

Problem 6. Let \(\displaystyle f,g: \RR\longrightarrow \RR\) be continuous functions such that \(\displaystyle g\) is differentiable. Assume that \(\displaystyle \bigl(f(0)-g^{\prime} (0)\bigr)\bigl(g^{\prime} (1)-f(1)\bigr)>0 \). Show that there exists a point \(\displaystyle c\in(0,1)\) such that \(\displaystyle f(c)=g^\prime (c)\).

Proposed by Fereshteh Malek, K. N. Toosi University of Technology

Hint: Apply the Mean Value Theorem (Darboux property) of derivatives.

  Solution