Question

If f(x) = [sin x] + [cos x], $x\in [0,2\pi ]$, where [.] denotes the greatest integer function. Then, the total number of points, where f(x) is non – differentiable, is

• A. 2
• B. 4
• C. 5
• D. 4

Answer 1

The correct option is C 5

Since, $\left[\sin x\right]=0$ $,x\in [0,\frac{\pi }{2})$
$=1$, $x=\frac{\pi }{2}$
$=0$, $x\in (\frac{\pi }{2},\pi ]$
$=-1$, $x\in (\pi ,2\pi )$
$=0$, $x=2\pi$
Similar way, $\left[\cos x\right]=1$ $,x=0$

$=0$, $x\in (0,\frac{\pi }{2}]$

$=-1$, $x\in (\frac{\pi }{2},\frac{3\pi }{2})$

$=0$, $x\in (\frac{3\pi }{2},2\pi )$

$=1$, $x=2\pi$
Thus, by adding above above two functions, we get
$f\left(x\right)=\left[\sin x\right]+\left[\cos x\right]=1$ $,x=0$

$=0$, $x\in (0,\frac{\pi }{2})$

$=1$, $x=\frac{\pi }{2}$

$=-1$, $x\in (\frac{\pi }{2},\pi ]$

$=-2$, $x\in (\pi ,\frac{3\pi }{2}]$

$=-1$, $x\in (\frac{3\pi }{2},2\pi )$

$=1$, $x=2\pi$

Therefore, the given function has $5$ discontinuity points over the given interval.
Hence, the given function has $5$ non-differentiable points.