Question

Let $f\left(x\right)=\sin x-x\cos x,x\in R$

The set of all values of $x\in (0,2\pi )$, for which $f\left(x\right)>0$ is

• A. $(0,\pi )$
• B. $(\pi ,2\pi )$
• C. $(\frac{\pi }{2},\frac{3\pi }{2})$
• D. None of these

Answer 1

The correct option is D None of these

For $x\in (0,\frac{\pi }{2}),f\left(x\right)>0$.
For $x\in (\frac{\pi }{2},\pi ),\sin x$ is positive, while $x\cos x$ is
negative, hence, here also $f\left(x\right)>0$.
If $\alpha$ is the least positive value for which $\tan x=x$, then $f\left(x\right)>0$ for $x\in (\pi ,\alpha )$.
For $x\in (\alpha ,2\pi ),f\left(x\right)\le 0$.
Thus, required set is $(0,\alpha )$.