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:

  1. 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.



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