Question

If $f\left(x\right)$ is a non-zero polynomial of degree four, having local extreme points at $x=-2,0,2$; then the set $S=\{x\in R:f(x)=f(0\left)\right\}$ contains exactly

• A. two irrational and two rational numbers
• B. two irrational and one rational numbers
• C. four rational numbers
• D. four irrational numbers

Answer 1

The correct option is B two irrational and one rational numbers

$f\left(x\right)=a_0{x}^4+a_1{x}^3+a_2{x}^2+a_{3}x+a_4$
$f^{\prime }\left(x\right)=4a_0{x}^3+3a_1{x}^2+2a_{2}x+a_3$

$f^{\prime }(-2)=-32a_0+12a_1-4a_2+a_3=0$
$f^{\prime }\left(0\right)=a_3=0$
$f^{\prime }\left(2\right)=32a_0+12a_1+4a_2+a_3=0$
$\Rightarrow a_1=a_3=0$

$32a_0+0+4a_2+0=0$
$\Rightarrow 32a_0=-4a_2\Rightarrow 8a_0=-a_2$

$\therefore f\left(x\right)=a_0{x}^4-8a_0{x}^2+a_4$
$f\left(0\right)=a_4$
$\therefore a_0{x}^4-8a_0{x}^2+a_4=a_4$
$\Rightarrow x^4-8x^2=0$
$\Rightarrow x^2(x^2-8)=0$
$\Rightarrow x=0,0,\sqrt{8},-\sqrt{8}$