Question:
If $f(\mathrm{x})$ is a non-zero polynomial of degree four, having local extreme points at $x=-1,0,1$; then the set $\mathrm{S}=\{\mathrm{x} \in \mathrm{R}: f(\mathrm{x})=f(0)\}$
Contains exactly :
Correct Option: , 2
Solution:
$f^{\prime}(x)=\lambda(x+1)(x-0)(x-1)=\lambda\left(x^{3}-x\right)$
$\Rightarrow f(x)=\lambda\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right)+\mu$
Now $f(\mathrm{x})=f(0)$
$\Rightarrow \lambda\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right)+\mu=\mu$
$\Rightarrow x=0,0, \pm \sqrt{2}$
Two irrational and one rational number