Question

Find the number of points where $f\left(x\right)=[\sin x+\cos x]$ (where $\left[\right]$ denotes the greatest integer function), $x\in (0,2\pi )$

• A. $5$
• B. $7$
• C. $10$
• D. $6$

Answer 1

The correct option is A $5$

Consider the given function,

$f\left(x\right)=[\sin x+\cos x].......\left(1\right)$

Where $\left[\right]$ denotes the greatest integer function.

Then,

Multiplying and divided by $\sqrt{2}$ in equation (1) and we get,

$f\left(x\right)=\sqrt{2}\left[\frac{\sin x+\cos x}{\sqrt{2}}\right]$

$f\left(x\right)=\sqrt{2}[\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x]$

$f\left(x\right)=\sqrt{2}[\sin x\cos \frac{\pi }{4}+\cos x\sin \frac{\pi }{4}]$

$f\left(x\right)=\sqrt{2}\left[\sin (x+\frac{\pi }{4})\right]$

So this function $f\left(x\right)=\sqrt{2}\left[\sin (x+\frac{\pi }{4})\right]$ will be discontinuous at the integral value in the interval $(0,2\pi )$.

Now, by inspection, we find that,

At $x=\frac{\pi }{2},\frac{3\pi }{4},\pi ,\frac{3\pi }{2},\frac{7\pi }{4}$

The value of $\sqrt{2}\sin (x+\frac{\pi }{4})$ is an integer

So there are $5$ points in the interval $(0,2\pi )$, for which $[\sin x+\cos x]$ is discontinuous

Hence, this is the answer.