CN-SMO 2006 Problem 1

Suppose $a>b>0$ , $f(x)=\dfrac{2(a+b)x+2ab}{4x+a+b}$ . Show that there exists an unique positive number $x$ , such that $f(x)=\left(\dfrac{a^{\frac{1}{3}}+b^{\frac{1}{3}}}{2} \right)^3$ .