Question 194
Suppose that $\alpha$ and $\beta$ are different real roots of the equation $4x^2 - 4tx - 1 = 0$ ($t \in \mathbb{R}$). $[\alpha, \beta]$ is the domain of the function $f(x) = \frac{2x-t}{x^2+1}$.
(1) Find $g(t) = \max f(x) - \min f(x)$.
(2) Prove that for $u_i \in (0, \frac{\pi}{2})$ ($i = 1, 2, 3$), if $\sin u_1 + \sin u_2 + \sin u_3 = 1$, then:
$$
\frac{1}{g(\tan u_1)} + \frac{1}{g(\tan u_2)} + \frac{1}{g(\tan u_3)} \lt \frac{3}{4}\sqrt{6}.
$$
(1) Find $g(t) = \max f(x) - \min f(x)$.
(2) Prove that for $u_i \in (0, \frac{\pi}{2})$ ($i = 1, 2, 3$), if $\sin u_1 + \sin u_2 + \sin u_3 = 1$, then:
$$
\frac{1}{g(\tan u_1)} + \frac{1}{g(\tan u_2)} + \frac{1}{g(\tan u_3)} \lt \frac{3}{4}\sqrt{6}.
$$