Question 62
Consider the function
$$
f(x) = -x + \sqrt{(x + a)(x + b)}
$$
where $a$ and $b$ are two given distinct positive real numbers.
Prove that for every real number $s$ belonging to the interval $(0,1)$ there exists a unique real number $\alpha$ such that
$$
f(\alpha) = \left( \frac{a^s + b^s}{2} \right)^{\frac{1}{s}}
$$
$$
f(x) = -x + \sqrt{(x + a)(x + b)}
$$
where $a$ and $b$ are two given distinct positive real numbers.
Prove that for every real number $s$ belonging to the interval $(0,1)$ there exists a unique real number $\alpha$ such that
$$
f(\alpha) = \left( \frac{a^s + b^s}{2} \right)^{\frac{1}{s}}
$$