Question

Let $f\left(x\right)=-x^{2}+x+p,$ where $p$ is a real number. If $g\left(x\right)=\left[f \left(x\right)\right]$ and $g\left(x\right)$ is discontinuous at $x=\frac{1}{2},$ then $p$ cannot be (where $\left[.\right]$ represents the greatest integer function)

• A. $\frac{1}{2}$
• B. $\frac{3}{4}$
• C. $\frac{7}{4}$
• D. $-\frac{1}{4}$

Answer 1

Answer: (A)

$f\left(x\right)=-\left(x - \frac{1}{2}\right)^{2}+\left(p + \frac{1}{4}\right)$
If $g\left(x\right)$ is discontinuous at $x=\frac{1}{2},$ then $f\left(\frac{1}{2}\right)$ should be an integer.
Hence, $p$ cannot be $\frac{1}{2}$