Question

Let $f(x)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\underset{x\to 0}{\lim }$$\left[1+\frac{f\left(x\right)}{x^2}\right]=3,$ then $f(2)$ is equal to

• A. $-8$
• B. $-4$
• C. $0$
• D. $4$

Answer 1

Answer: (C)

$f(x)=$
$\underset{x\to 0}{\lim }\left[1+\frac{f(x)}{x^2}\right]=3$
$\Rightarrow f(x)$ must not contain degree $0\&$ degree $1$ term
$\Rightarrow f(x)=a{x}^4+b{x}^3+c{x}^2$
now $f^{\prime }(x)=4a{x}^3+3b{x}^2+2cx$
$f^{\prime }(1)=4a+3b+2c=0\dots \dots (1)$
$f^{\prime }(2)=32a+12b+4c=0\dots \dots (2)$
and $\underset{x\to 0}{\lim }\left[1+\frac{f(x)}{x^2}\right]=1+c=3\dots \dots (3)$
$\Rightarrow c=2$