Question

Let $f\left(x\right)$ be a polynomial function satisfying the following conditions
$\underset{x\to \infty }{lim}\frac{f\left(x\right)}{|x{|}^3}=0,\underset{x\to \infty }{lim}(\sqrt{f\left(x\right)}-x)=-1andf\left(0\right)=0$
(where, [.] denotes greatest integer function) then which of the following is/are correct?

• A. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 3
• B. The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 5
• C. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 4
• D. The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 3

Answer 1

Answer: (B), (D)

The correct options are
B The number of points of discontinuity of $g\left(x\right)=\left[f\right(x\left)\right]$ in [0, 3] is 5
D The number of points of non-derivability of $h\left(x\right)=|f\left(|x|\right)|$ is 3

$\underset{x\to \infty }{lim}\frac{f\left(x\right)}{|x{|}^3}=0\Rightarrow f\left(x\right)$ is linear or quadratic and $f\left(0\right)=0$
So $f\left(x\right)=a{x}^2+bx$
$\underset{x\to \infty }{lim}(\sqrt{a{x}^2+bx}-x)=\underset{x\to \infty }{lim}\frac{(a-1)x^2+bx}{x(\sqrt{a+\frac{b}{x}}+1)}=-1\Rightarrow \underset{x\to \infty }{lim}\frac{(a-1)x+b}{(\sqrt{a+\frac{b}{x}}+1)}=-1$
$\therefore a-1=0and\frac{b}{\sqrt{a}+1}=-1\Rightarrow b=-2$
So $f\left(x\right)=x^2-2x$
$g\left(x\right)=[x^2-2x],x\in [0,3]$


$\therefore g\left(x\right)=\left[f\right(x\left)\right]$ has total 5 points of discontinuity.
$h\left(x\right)=|f\left(|x|\right)|=||x{|}^2-2|x||$

$\therefore h\left(x\right)=|f\left(|x|\right)|$ has total 3 points of non-derivability.