Question

If $f\left(x\right)=\sqrt{{\log }_{10}\left(\frac{2\left[x\right]+20}{[x{]}^2-4}\right)}$, where $[.]$ denotes the greatest integer function and $S_1=\left\{x|f\right(x)\text{is defined};x\in I^{+}\}{S}_2=\{N|S_2\subset S_1;N\in I^{+}\}$
Then the maximum degree of polynomial having distinct roots $\left\{S_2\right\}$ is

Answer 1

For $f\left(x\right)$ to be defined
$\frac{2\left[x\right]+20}{[x{]}^2-4}\ge 1\&[x{]}^2-4\ne 0\Rightarrow [x]\ne 2,-2\frac{2\left[x\right]+20}{[x{]}^2-4}\ge 1\Rightarrow \frac{-[x{]}^2+2[x]+24}{[x{]}^2-4}\ge 0\Rightarrow \frac{\left(\right[x]-6)(-[x]-4)}{\left(\right[x]-2)\left(\right[x]+2)}\ge 0\Rightarrow \frac{\left(\right[x]-6)\left(\right[x]+4)}{\left(\right[x]-2)\left(\right[x]+2)}\le 0S_1\text{is}[-4,-2)\cup [3,7)S_2=\{3,4,5,6\}$
So, total root of the polynomial =$4$
So, maximum degree is $4.$