Application of Derivatives 4 Question 4
####4. If $f(x)$ 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
(a) four rational numbers
(2019 Main, 9 April I)
(b) two irrational and two rational numbers
(c) four irrational numbers
(d) two irrational and one rational number
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Answer:
Correct Answer: 4. (d)
Solution:
- The non-zero four degree polynomial $f(x)$ has extremum points at $x=-1,0,1$, so we can assume $f^{\prime}(x)=a(x+1)(x-0)(x-1)=a x\left(x^{2}-1\right)$
where, $a$ is non-zero constant.
$ f^{\prime}(x)=a x^{3}-a x $
$\Rightarrow \quad f(x)=\frac{a}{4} x^{4}-\frac{a}{2} x^{2}+C$
[integrating both sides]
where, $C$ is constant of integration.
Now, since $f(x)=f(0)$
$\Rightarrow \frac{a}{4} x^{4}-\frac{a}{2} x^{2}+C=C \Rightarrow \frac{x^{4}}{4}=\frac{x^{2}}{2}$
$ \Rightarrow \quad x^{2}\left(x^{2}-2\right)=0 \Rightarrow x=-\sqrt{2}, 0, \sqrt{2} $
Thus, $f(x)=f(0)$ has one rational and two irrational roots.
