a non-zero polynomial of degree four, having local extreme points at

Question:

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

  1. (1) four irrational numbers.

  2. (2) four rational numbers.

  3. (3) two irrational and two rational numbers.

  4. (4) two irrational and one rational number.


Correct Option: 4,

Solution:

Since, function $\mathrm{f}(\mathrm{x})$ have local extreem points at $\mathrm{x}=$

$-1,0,1$. Then

$f(x)=K(x+1) x(x-1)$

$=\mathrm{K}\left(\mathrm{x}^{3}-\mathrm{x}\right)$

$\Rightarrow \mathrm{f}(\mathrm{x})=K\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right)+C$ (using integration)

$\Rightarrow \mathrm{f}(0)=\mathrm{C}$

$\because \mathrm{f}(\mathrm{x})=\mathrm{f}(0) \Rightarrow K\left(\frac{x^{4}}{4}-\frac{x^{2}}{2}\right)=0$

$\Rightarrow \frac{x^{2}}{2}\left(\frac{x^{2}}{2}-1\right)=0$

$\Rightarrow x=0,0, \sqrt{2},-\sqrt{2}$

$\therefore S=\{0,-\sqrt{2}, \sqrt{2}\}$

 

 

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