Question

If $f\left(x\right)$ is a non-zero polynomial of degree 4 , 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. two irrational and two rational numbers.
• B. two irrational and one rational number.
• C. four rational numbers.
• D. four irrational numbers.

Answer 1

The correct option is B two irrational and one rational number.

Given $f\left(x\right)$ has extremes at $x=-1,0$ and $1$.
$\Rightarrow f^{\prime }\left(x\right)$ is zero at $x=-1,0and1$.
$\Rightarrow f^{\prime }\left(x\right)=k(x-1)x(x+1)=k(x^3-x)\Rightarrow f\left(x\right)=k\cdot [\frac{x^4}{4}-\frac{x^2}{2}]+qNowf\left(x\right)=f\left(0\right)\Rightarrow k\cdot [\frac{x^4}{4}-\frac{x^2}{2}]+q=q\Rightarrow k\cdot x^2(x^2-2)=0\Rightarrow x=0,0,\pm \sqrt{2}$
$\therefore$ two irrational numbers and 1 rational number.