Question

Let $f(x)$ be a polynomial of degree $5$ such that $x=\pm 1$ are its critical points. If $\left(2+\frac{f\left(x\right)}{x^3}\right)=4$ then which one of the following is not true ?

• A. $f\left(1\right)-4f\left(-1\right)=4$
• B. $x=1$ is a point of maxima and $x=-1$ is a point of minimum of $f$
• C. $f$ is an odd function
• D. $x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$

Answer 1

Answer: (D)

$\underset{x\to 0}{\lim }$$\left(2+\frac{f\left(x\right)}{x^3}\right)=4$
$\Rightarrow f\left(x\right)=2x^3+a{x}^4+b{x}^5$
$f^{\prime }\left(x\right)=6x^2+4a{x}^3+5b{x}^4$
$f^{\prime }\left(1\right)=0,f^{\prime }\left(-1\right)=0$
$a=0,b=\frac{-6}{5}\Rightarrow f\left(x\right)=2x^3-\frac{6}{5}{x}^5$
$f^{\prime }\left(x\right)=6x^2-6x^4$
$=6x^2\left(1-x\right)\left(1+x\right)$
Sign scheme for $f^{\prime }\left(x\right)$
Minima at $x=-1$
Minima at $x=1$