Question

If range of the function $f ( x )=\sin ^{-1} x -\cot ^{-1} x + x ^2+2 x +6$ is $[ a , b ]$ then $[ a + b ]$ equals
[Note : [k] denotes greatest integer less than or equal to $k$.]

• A. 5
• B. 6
• C. 10
• D. 11

Answer 1

Answer: (C)

$f(x)=\sin ^{-1} x-\left(\frac{\pi}{2}-\tan ^{-1} x\right)+(x+1)^2+5=\sin ^{-1} x+\tan ^{-1} x-\frac{\pi}{2}+(x+1)^2+5$
Domain of $f(x)$ is $[-1,1]$ and $f(x)$ is increasing in $[-1,1]$
$f (-1)=\frac{-5 \pi}{4}+5, f (1)=\frac{\pi}{4}+9 $
$\therefore \text { Range of } f ( x ) \text { is }\left[\frac{-5 \pi}{4}+5, \frac{\pi}{4}+9\right]$
$\Rightarrow a + b =-\pi+14 \Rightarrow[ a + b ]=10$