Question

Let $f( x )=\sin x +\cos x +\tan x +\arcsin x +\arccos x +\operatorname{arc} \tan x$. If $M$ and $m$ are maximum and minimum values of $f( x )$ then their arithmetic mean is equal to

• A. $ \frac{\pi}{2}+\cos 1$
• B. $\frac{\pi}{2}+\sin 1$
• C. $\frac{\pi}{4}+\tan 1+\cos 1$
• D. $\frac{\pi}{4}+\tan 1+\sin 1$

Answer 1

Answer: (A)

Domain of $f$ is $[-1,1] ; f ( x )=\sin x +\cos x +\tan x +\sin ^{-1} x +\cos ^{-1} x +\tan ^{-1} x$

$\text { Hence } f ^{\prime}( x )>0 \Rightarrow \text { fisincreasing } \Rightarrow \text { range is }[ f (-1), f (1)] $
$\left.\therefore f ( x )\right|_{\min }= f (-1)=-\sin 1+\cos 1-\tan 1-\frac{\pi}{2}+\pi-\frac{\pi}{4}=\frac{\pi}{4}+\cos 1-\sin 1-\tan 1 $
$\left. f ( x )\right|_{\max }= f (1)=\sin 1+\cos 1+\tan 1+\frac{\pi}{2}+\frac{\pi}{4}=\frac{3 \pi}{4}+\cos 1+\sin 1+\tan 1 $
$\Rightarrow \frac{ M + m }{2}=\frac{\pi}{2}+\cos 1 \Rightarrow \text { (A) }$