If f ( x ) is a non-zero polynomial of degree four, having local extreme points at x =   – 1 ,…

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∈R :fx=f0} contains exactly
  1. Two irrational and two rational numbers
  2. Four rational numbers
  3. Two irrational and one rational number
  4. Four irrational numbers

Solution

f(x) has extreme points at {-1, 0, 1}

⇒f'x=0  at x= –1, 0, 1

⇒f'x=ax2-1x  ∵f x is a polynomial

⇒fx=∫ax3-xdx=ax44-x22+ C

Now the equation,

fx=f(0)

⇒ax44-x22+ C=C
⇒ x44-x22=0 ⇒x2x2-2=0
⇒x=0, ±2

⇒S= {0, 2, -2}.

Hence, the set contains two irrational and one rational number.

Asked in: JEE Main 2019 (09 Apr Shift 1)

Practice more Applications of Derivatives questions on Aicharya