Question

Let $f(x)$ be a polynomial of degree $5$ such that $x = ±1$ 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)

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