Question

If range of the function $f(x)=\sin ^{-1} x+2 \tan ^{-1} x+x^2+4 x+1$ is [p, q] then find the value of $(p+q)$

Answer 1

We have $f(x)=\sin ^{-1} x+2 \tan ^{-1} x+x^2+4 x+1$
Clearly domain of $f(x)$ is $[-1,1]$.
Also $f(x)$ is increasing function in its domain.
$\therefore p = f _{\min .}( x )= f (-1)=-\frac{\pi}{2}+2\left(\frac{-\pi}{4}\right)+1-4+1=-\pi-2 . $
$q = f _{\max .}( x )= f (1)=\frac{\pi}{2}+2\left(\frac{\pi}{4}\right)+1+4+1=\pi+6$
$\therefore \text { Range of } f ( x ) \text { is }[-\pi-2, \pi+6] $
$\text { Hence }( p + q )=4$
$\text { Note }: \text { Vertex of } y = x ^2+4 x +1 \text { is at } x =-2 \text { and hence in the domain }\left( x ^2+4 x +1\right) \text { is increasing.] }$