IMC 2019 Problem 6

Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$ . Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$ .

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