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Solve this following

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 :

1. four irrational numbers.

2. two irrational and one rational number.

3. four rational numbers.

4. two irrational and two rational numbes.

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