Question

Find the values of $x$ satisfying $\left[x\right]-1+x^2\ge 0;$ where $[.]$ denotes the greatest integer function.

• A. $x\in (-\infty ,-\sqrt{3}]\cup [1,\infty )$
• B. $x\in (-\infty ,-\sqrt{2}]\cup [2,\infty )$
• C. $x\in (-\infty ,-3]\cup [3,\infty )$
• D. $x\in (-\infty ,-\sqrt{3}]\cup [3,\infty )$

Answer 1

The correct option is A $x\in (-\infty ,-\sqrt{3}]\cup [1,\infty )$

As, $\left[x\right]-1+x^2\ge 0;$
$\Rightarrow$ $x^2-1\ge -\left[x\right]$
Thus, to find the points for which $f\left(x\right)=x^2-1$ is greater than or equal to $g\left(x\right)=-\left[x\right]$,
where the two functions $f\left(x\right)$ and $g\left(x\right)$ can be plotted as shown in the graph.
Thus, from the graph $f\left(x\right)\ge g\left(x\right)$ when $x\in (-\infty ,A]\cup [B,\infty )$ where $A$ is point of intersection of $x^2-1$ and $-\left[x\right]=+2$
$\therefore x^2-1=+2$ ie, $x=-\sqrt{3}=A$ and $B=1$
$\therefore \left[x\right]-1+x^2\ge 0$ is satisfied, $\forall x\in (-\infty ,-\sqrt{3}]\cup [1,\infty )$