Question

Let $f(x)$ be the remainder when $p(x)=x^{73}-2 x^6+3 x^3$ is divided by $x^4\left(x^2-1\right)$. Identify which of the following statement is (are) correct?

• A. Range of $f(f(x))$ is $R$
• B. Range of $f ( f ( x ))$ is $[0, \infty)$
• C. $f ( x )$ has exactly 3 points of inflection
• D. $f(x)$ has exactly one point of inflection

Answer 1

Answer: (A), (D)

$P(x)=x^{73}-2 x^6+3 x^3 $
$P(x)=Q(x) \cdot x^4 \cdot\left(x^2-1\right)+\left(a x^2+b x+c\right) x^3$
$\text { After cancelling } x^3 $
$\text { Put } x=0 \Rightarrow c=3 $
$\text { Put } x=-1 \Rightarrow a+b+c=2 $
$\Rightarrow a=1, b=-2, c=3 $
$f(x)=x^5-2 x^4+3 x^3 $
$\text { Degree of } f ( f ( x ))=25 \rightarrow \text { odd }$
$\Rightarrow \text { Range } \in(-\infty, \infty) $
$f ^{\prime}( x )= x ^2\left(5 x ^2-8 x +9\right)$
$f ^{\prime \prime}( x )=20 x ^3-24 x ^2+18 x$
Only one real root.