Question

If $f\left(x\right)$ is a non-zero polynomial of degree four, having local extreme points at $x=-1,0,1$; then the set $S=\{x\in R:f(x)=f(0\left)\right\}$
Contains exactly:

• A. Four irrational numbers
• B. Two irrational and one rational number
• C. Four rational numbers
• D. Two irrational and two rational numbers

Answer 1

The correct option is B Two irrational and one rational number

$f^{\prime }\left(x\right)=\lambda (x+1)(x-0)(x-1)=\lambda (x^3-x)$
$\Rightarrow f\left(x\right)=\lambda (\frac{x^4}{4}-\frac{x^2}{2})+\mu$
Now $f\left(x\right)=f\left(0\right)$
$\Rightarrow \lambda (\frac{x^4}{4}-\frac{x^2}{2})+\mu =\mu$
$\Rightarrow x=0,0,\pm \sqrt{2}$
Two irrational and one rational number.