QJEEM19Maa654: JEE Main 2019 Mathematics - Class 12 JEEM19Maa654 Solution: If ƒ(x) is a non-zero polynomial of degree four, having local…

Question

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If ƒ(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 : ƒ(x) = ƒ(0)}
Contains exactly :

Accepted Answer

Local extreme points of f(x) is at x = –1, 0, 1.

\therefore

f'(x) = 0 has three solutions x = –1, 0, 1.

\therefore

f'(x) = k(x + 1)x(x - 1)

\int {f'(x)dx} = \int {k(x + 1)x(x - 1)dx}

f(x) =

k\left[ {{{{x^4}} \over 4} - {{{x^2}} \over 2}} \right] + C

Also given that

ƒ(x) = ƒ(0)

\therefore

k\left[ {{{{x^4}} \over 4} - {{{x^2}} \over 2}} \right] + C

= C

\Rightarrow

{{{x^2}} \over 2}\left( {{{{x^2}} \over 2} - 1} \right)

= 0

\Rightarrow

x = 0,

- \sqrt 2

,

\sqrt 2

\therefore

x has two irrational and one rational value.

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