JEE Main · 2019 · 09 Apr 2019 · Shift 1 · Application of Derivatives

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:

Answer: (C)

Solution

With local extrema at

x = - 1, 0, 1

, the derivative is

f^{'} (x) = λ x (x + 1) (x - 1) = λ (x^{3} - x)

, so

f (x) = λ (\frac{x ^{4}}{4} - \frac{x ^{2}}{2}) + μ

and

f (0) = μ

. Solving

f (x) = f (0)

means

\frac{x ^{4}}{4} - \frac{x ^{2}}{2} = 0

, i.e.

x^{2} (x^{2} - 2) = 0

, with roots

x = 0

and

x = \pm 2

. So the set holds one rational number,

0

, together with the two irrationals

\pm 2

JEE Main · 2019 · 09 Apr 2019 · Shift 1 · Single correct option