VMO 2006 Problem 4

Given is the function $f(x)=-x+\sqrt{(x+a)(x+b)}$ , where $a$ , $b$ are distinct given positive real numbers. Prove that for all real numbers $s\in (0,1)$ there exist only one positive real number $\alpha$ such that $$f(\alpha)=\sqrt [s]{\frac{a^s+b^s}{2}} .$$