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: f(x) = f(0)} contains exactly:

The correct option is B Two irrational and one rational number
f'(x) = λ(x+1)(x-0)(x-1) = λ(x³ - x) ⇒ f(x) = λ(x⁴/4 - x²/2) + μ
Now f(x) = f(0) ⇒ λ(x⁴/4 - x²/2) + μ = μ ⇒ x⁴/4 - x²/2 = 0 ⇒ x²(x² - 2) = 0
Thus x = 0, 0, ±√2
Two irrational and one rational number.