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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:

1. four irrational numbers.
2. four rational numbers.
3. two irrational and two rational numbers.
4. two irrational and one rational number.

Correct answer: two irrational and one rational number.

Solution

Since f(x) is a degree four polynomial, it can have at most four real roots. Given that it has local extrema at x = -1, 0, and 1, the function must cross the horizontal line y = f(0) at least three times, leading to two additional roots. This results in a total of three distinct points where f(x) equals f(0), which can include two irrational and one rational number.

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