RIOPLATENSE 2016 Problem 4

Let $c > 1$ be a real number. A function $f: [0 ,1 ] \to R$ is called c-friendly if $f(0) = 0, f(1) = 1$ and $|f(x) -f(y)| \le c|x - y|$ for all the numbers $x ,y \in [0,1]$ . Find the maximum of the expression $|f(x) - f(y)|$ for all c-friendly functions $f$ and for all the numbers $x,y \in [0,1]$ .