SATHEE: Application of Derivatives 4 Question 4

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

(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 $f^{'} (x) = a (x + 1) (x - 0) (x - 1) = a x (x^{2} - 1)$

where, $a$ is non-zero constant.

$f^{'} (x) = a x^{3} - a x$

$\Rightarrow f (x) = \frac{a}{4} x^{4} - \frac{a}{2} x^{2} + C$

[integrating both sides]

where, $C$ is constant of integration.

Now, since $f (x) = f (0)$

$\Rightarrow \frac{a}{4} x^{4} - \frac{a}{2} x^{2} + C = C \Rightarrow \frac{x^{4}}{4} = \frac{x^{2}}{2}$

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

Thus, $f (x) = f (0)$ has one rational and two irrational roots.

Application of Derivatives 4 Question 4

Answer:

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Application of Derivatives 4 Question 4

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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