Question 4
Let $f(x)=-x+\sqrt{(a+x)(b+x)}$, where $a,b$ are given distinct positive real numbers. Prove that for each $s\in(0,1)$ there exists a unique positive real number $\alpha$ such that
$$f(\alpha)=\left(\frac{a^s+b^s}{2}\right)^{1/s}.$$
$$f(\alpha)=\left(\frac{a^s+b^s}{2}\right)^{1/s}.$$