Question

Let $f(x)=[3+4sinx]$ (where [ ] denotes the greatest integer function). If sum of all the values of $x$ in $[\pi ,2\pi ]$, where $f(x)$ fails to be differentiable, is $\frac{k\pi }{2}$, then the value of $k$ is

Answer 1

Answer: 24

$f(x)=[3+4sinx]=3+[4sinx]$
The graph of $y=3+[4sinx]$ in $[\pi ,2\pi ]$ is as shown


discontinuous and hence non-derivable at $8$ points which are
$\pi ,\pi +si{n}^{-1}(\frac{1}{4}),\pi +si{n}^{-1}(\frac{2}{4}),\pi +si{n}^{-1}(\frac{3}{4})$,
$2\pi -si{n}^{-1}(\frac{1}{4}),2\pi -si{n}^{-1}(\frac{2}{4}),2\pi -si{n}^{-1}(\frac{3}{4}),2\pi$
$\Rightarrow$ sum $=12\pi =\frac{k\pi }{2}$
$\Rightarrow k=24$