Let f(x)=-x2+x+p, where p is a real number. If g(x)=[f (x)] and g(x) is discontinuous at x=(1/2), then p cannot be (where [.] represents the greatest integer function)

Question

Let f(x)=-x2+x+p, where p is a real number. If g(x)=[f (x)] and g(x) is discontinuous at x=(1/2), then p cannot be (where [.] represents the greatest integer function) (A) (1/2) (B) (3/4) (C) (7/4) (D) -(1/4)

Tardigrade Admin · Accepted Answer

f(x)=-(x - (1/2))2+(p + (1/4)) If g(x) is discontinuous at x=(1/2), then f((1/2)) should be an integer. Hence, p cannot be (1/2)

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

$\frac{1}{2}$

$\frac{3}{4}$

$\frac{7}{4}$

$- \frac{1}{4}$

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

Q. Let f(x)=−x2+x+p, where p is a real number. If g(x)=[f(x)] and g(x) is discontinuous at x=21​, then p cannot be (where [.] represents the greatest integer function)

21​

43​

47​

−41​