Question

Let f(x)=$\frac{\sin 4\pi \left[x\right]}{1+{\left[x\right]}^2}$, where $\left[x\right]$ greatest integer less than or equal to $x$, then

• A. $f\left(x\right)$ is not diffentiable at some points
• B. $f\left(x\right)$ exists but in different from zero
• C. $LHD\left(atx=0\right)=0,RHD\left(atx=1\right)=0$
• D. $f\left(x\right)$ but F is not a constant function.

Answer 1

The correct option is B $f\left(x\right)$ exists but in different from zero

$f\left(x\right)=\frac{\sin 4\pi \left[x\right]}{1+[x{]}^2}$

$\sin (4\pi \times Integer)=\sin \left(4n\pi \right)=0$

$f\left(x\right)=0$

$1+[x{]}^2$ is always positive, So denominator can't be $0$

$LHD(x=0)=0$

$RHD(x=1)=0$

$B$ is correct