Question

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

• A. f(x) is not differentiable at some points
• B. f(x) exists but is different from zero
• C. f(x)=0 for all x
• D. f' (x) =0 but f is not a constant function

Answer 1

Answer: (C)

f(x)=0 for all x

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

We know that $\left[x\right]$ given us an integer value

for any x

$\sin 4\pi \left[x\right]=\sin 4\pi n\left(nϵI\right)$ for any x

$=0$

$1+{\left[x\right]}^2\ge 1$

$\Rightarrow f\left(x\right)=\frac{0}{1+{\left[x\right]}^2}=0$ for all x