Question

Consider $f\left(x\right)=\{\begin{array}{c}\left[\frac{2(\sin x-{\sin }^{3}x)+|\sin x-{\sin }^{3}x|}{2(\sin x-{\sin }^{3}x)-|\sin x-{\sin }^{3}x|}\right],x\ne \frac{\pi }{2}forx\in (0,\pi )\\ 3x=\frac{\pi }{2}\end{array}$, where $\left[\right]$ denotes the greatest integer function, then-

• A. $f$ is continuous and differentiable at $x=\pi /2$
• B. $f$ is continuous but not differentiable at $x=\pi /2$
• C. $f$ is neither continuous not differentiable at $x=\pi /2$
• D. None of these

Answer 1

The correct option is A $f$ is continuous and differentiable at $x=\pi /2$

$f\left(x\right)=\{\begin{array}{c}\left[\frac{2(\sin x-{\sin }^{3}x)+(\sin x-{\sin }^{3}x)}{2(\sin x-{\sin }^{3}x)-(\sin x-{\sin }^{3}x)}\right],x\ne \frac{\pi }{2}forx\in (0,\pi )\\ 3x=\frac{\pi }{2}\end{array}$ ($\because \sin x>{\sin }^{3}x$ in $(0,\pi )$]
Now $\{\begin{array}{c}f\left(x\right)=3;x\ne \frac{\pi }{2}\\ =3;x=\frac{\pi }{2}\end{array}$
Hence $f\left(x\right)$ is continuous and differentiable at $x=\frac{\pi }{2}$