Question

Let $f(x)=\sin x+\cos x+\tan x+\arcsin x+\arccos x+\mathrm{arc}\tan x$. 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 $[-1,1];f(x)=\sin x+\cos x+\tan x+{\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x$

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