Question

For a real number x let $\left[x\right]$ denote the largest number less than or equal to x. for $xϵR$ let $f\left(x\right)=\left[x\right]\sin \pi x$. Then.

• A. f is differentiable on R
• B. f is symmetric about the line $x=0$
• C. ${\int }_{-3}^{3}f\left(x\right)dx=0$
• D. For each real $\alpha$, the equation $f\left(x\right)-\alpha =0$ has infinitely many roots.

Answer 1

The correct option is D For each real $\alpha$, the equation $f\left(x\right)-\alpha =0$ has infinitely many roots.

$f\left(x\right)=\left[x\right]\sin \pi x$

$f\left(x\right)=\{\begin{array}{c}-sin\pi xx\in [-1,0)\\ 0x\in [0,1)\\ sin\pi xx\in [1,2)\end{array}$

Similarly $f\left(x\right)$ will be defined for all other intervals

$\sin \pi x$ is always continous

From the graph of the function

$f\left(x\right)$ is not differntiable at $x=1$. So, $\left(A\right)$ is false

$f\left(x\right)$ is not symemetric about $x=0$. So, $\left(B\right)$ and $\left(C\right)$ are false

If we draw any line $f\left(x\right)=\alpha$ this will cut the curve at infinite points

So option $\left(D\right)$ is correct.