Question

If $f\left(x\right)=\left[x\right]-1+x^2$, where $[\cdot ]$ represents greatest integer function, then which of the following is/are correct

• A. $f\left(x\right)=0$ has $2$ solutions.
• B. The sum of the solution(s) for $f\left(x\right)=0$ is $0$
• C. $f\left(x\right)=0$ has $1$ solution.
• D. The sum of the solution(s) $f\left(x\right)=0$ is $-\sqrt{3}$

Answer 1

Answer: (C), (D)

The sum of the solution(s) $f\left(x\right)=0$ is $-\sqrt{3}$

The number of solution(s) of the equation $\left[x\right]-1+x^2=0$ is equal to the number of point of intersection of the curves $y=\left[x\right]$ and $y=1-x^2,$ shown as


$\therefore$ There is only $1$ solution.

Clearly, $1-x^2$ and $\left[x\right]$ intersect when $\left[x\right]=-2$
$\Rightarrow 1-x^2=-2\Rightarrow x^2=3\Rightarrow x=\pm \sqrt{3}$
(rejecting $x=\sqrt{3}$ from graph)
$\Rightarrow x=-\sqrt{3}$ is the only soluion.
Hence, the sum of the solution(s) is $-\sqrt{3}$.