Question

Let $f(x)$ be the remainder when $p(x)=x^{73}-2x^6+3x^3$ 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)

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