Question

Let $f\left(x\right)=\frac{x^4-\lambda x^3-3x^2+3x\lambda }{x-\lambda },x\in R-\left\{\lambda \right\}$. If the range of $f\left(x\right)$ is $R$, then the complete set of values of $\lambda$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

• A. $[-2,2]$
• B. $[0,4]$
• C. $(1,3)$
• D. $(-2,2)$

Answer 1

The correct option is A $[-2,2]$

Given : $f\left(x\right)=\frac{x^4-\lambda x^3-3x^2+3x\lambda }{x-\lambda }$
$\Rightarrow f\left(x\right)=\frac{(x^3-3x)(x-\lambda )}{x-\lambda }\Rightarrow f\left(x\right)=\frac{x(x^2-3)(x-\lambda )}{x-\lambda }$

Considering $g\left(x\right)=x^3-3x$


$f\left(x\right)=g\left(x\right)$ except $x=\lambda$
If $\lambda >2$, then from the range of $f\left(x\right)$, $f\left(\lambda \right)$ is missed,
Similarly, if $\lambda <2$, then from the range of $f\left(x\right)$, $f\left(\lambda \right)$ is missed,
But when $\lambda \in [-2,2]$, $f\left(\lambda \right)$ is achieved at some other point than $x=\lambda$

Hence, $\lambda \in [-2,2]$