Question

Let $f(x) = [3 + 4 \,sin \,x]$ (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

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


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