Question

Let , $f\left(x\right)=[\cos x+\sin x],0<x<2\pi ,where\left[x\right]$ denotes the greatest integer less than or equal to $\left[x\right]$. The number of points of discontinuity of $f\left(x\right)$ is

• A. 6
• B. 5
• C. 4
• D. 3

Answer 1

Answer: (B)

5

we know ,[P] is discontinuances at integral value of P

Now, $sinx+cosx=\sqrt{2}sin(x+\pi /4)$

$[sinx+cosx]=\left[\sqrt{2}sin\right(x+\frac{\pi }{4}\left)\right]$ will be discontinues

at integral values of $\sqrt{2}sin(x+\pi /4)$ in the interval $(0,2\pi )$

$x=\pi /2,3\pi /4,\pi ,3\pi /2,7\pi /4,\sqrt{2}sin(x+\pi /4)$ is an integer

there are 5 point $(0,2\pi )$ to which $[sinx+cosx]$ is discontinues.