Question

Let $f(x)$ be a polynomial of degree four having extreme values at $x = 1$ and $x = 2$. If $\displaystyle \lim_{x \to 0}$$\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)

$f(x)=$
$\displaystyle\lim_{x \rightarrow 0}\left[1+\frac{f(x)}{x^{2}}\right]=3$
$\Rightarrow f(x)$ must not contain degree $0 \&$ degree $1$ term
$\Rightarrow f(x)=a x^{4}+b x^{3}+c x^{2} $
now $ f'(x)=4 a x^{3}+3 b x^{2}+2 c x $
$ f'(1)=4 a+3 b+2 c=0 \ldots \ldots(1) $
$f'(2)=32 a+12 b+4 c=0 \ldots \ldots(2)$
and $\displaystyle\lim_{x \rightarrow 0}\left[1+\frac{f(x)}{x^{2}}\right]=1+c=3 \ldots \ldots(3)$
$\Rightarrow c=2$