Question

Let $f\left(x\right)$ be a polynomial of degree $5$ such that $x=1,-1$ are its critical points. If $\underset{x\to 0}{\lim }\left(2+\frac{f\left(x\right)}{x^3}\right)=4$ then which 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

The correct option is D

$x=1$ is a point of minima and $x=-1$ is a point of maxima of $f$

Explanation for the correct option:

Option (C)

$$\begin{gathered} \underset{x\to 0}{\lim }\left(2+\frac{f\left(x\right)}{x^3}\right)=4 \\ \Rightarrow \underset{x\to 0}{\lim }\left(2\right)+\underset{x\to 0}{\lim }\left(\frac{f\left(x\right)}{x^3}\right)=4 \\ \Rightarrow \underset{x\to 0}{\lim }\left(\frac{f\left(x\right)}{x^3}\right)=2 \end{gathered}$$

so limit exists and is finite hence in $f\left(x\right)$ coefficient of $x^2,x,1$ will be $0$

let $f\left(x\right)=a{x}^5+b{x}^4+c{x}^3$

$\underset{x\to 0}{\lim }\left(\frac{a{x}^5+b{x}^4+c{x}^3}{x^3}\right)=c=2$

since $1,-1$are critical points

$f\text{'}\left(x\right)=5a{x}^4+4b{x}^3+3c{x}^2$

$$\begin{gathered} f\text{'}\left(1\right)=5a+4b+6=0 \\ f\text{'}(-1)=5a-4b+6=0 \end{gathered}$$

Solving the equations we get

$b=0,a=\frac{-6}{5}$

Therefore

$f\left(x\right)=\frac{-6}{5}{x}^5+2x^3$ which is an odd function

Option (B):

$f\left(x\right)=\frac{-6}{5}{x}^5+2x^3$

$$\begin{gathered} f\text{'}\left(x\right)=-6x^4+6x^2 \\ f\text{'}\text{'}\left(x\right)=-24x^3+12x \\ f\text{'}\text{'}\left(1\right)<0 \\ f\text{'}\text{'}(-1)>0 \end{gathered}$$

At $x=-1$ there is local minima and at $x=1$ there is local maxima.

Hence option (B) is true

Option (A):

$f\left(x\right)=\frac{-6}{5}{x}^5+2x^3$

$f\left(1\right)-4f(-1)=4$

Explanation for the incorrect option:

Option(D):

We know that At $x=-1$ there is local minima and at $x=1$ there is local maxima.

Hence option (D) is incorrect

And we had to find the incorrect statement

Hence, the correct option is (D)