Question

Let $f\left(x\right)=\sin x+\cos x+\tan x+\arcsin x+\arccos x+\arctan x$. 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

The correct option is A $\frac{\pi }{2}+\cos 1$

The domain of $f\left(x\right)$ will be $[-1,1]$
Hence
$f(-1)$
$=-\sin 1+\cos 1-\tan 1-\frac{\pi }{2}+\pi -\frac{\pi }{4}$
$=m$ ...(i)
$f\left(1\right)$
$=\sin 1+\cos 1+\tan 1+\frac{\pi }{2}+0+\frac{\pi }{4}$
$=M$ ...(ii)
Hence $M+m$
$=2\cos 1+\pi$
Thus mean will be
$\frac{m+M}{2}$ $=\cos 1+\frac{\pi }{2}$