If $\alpha ≤ sin^{-1}x +cos^{-1}x+tan^{-1}x ≤\beta $, then 

$\alpha = \frac{\pi}{4}, \beta =\frac{3\pi}{4}$

We have,

$sin^{-1}x +cos^{-1}x =\frac{\pi}{2}$ for all x ∈ [-1,1]

Also,

$-\frac{\pi}{4}≤tan x ≤\frac{\pi}{4}$ for all x ∈ [-1,1]

$∴ -\frac{\pi}{4}+\frac{\pi}{2} ≤sin^{-1}x +cos^{-1}x+tan^{-1}x≤\frac{\pi}{4}+\frac{\pi}{2}$

$⇒ \frac{\pi}{4}≤ sin^{-1}x +cos^{-1}x +tan^{-1}x ≤ \frac{3\pi}{4}$