Let a, b, c be non-zero real numbers such that b² < 4ac, and f(x) = ax² + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

Options

1. the set of all integers
2. either the empty set or the set of all integers
3. the empty set
4. the set of all positive integers

Correct Answer

either the empty set or the set of all integers

Explanation

Since b² - 4ac < 0, the quadratic f(x) = ax² + bx + c has no real roots, so its sign is constant for all real x. If a > 0, then f(x) > 0 for all x ⇒ S = ∅ (empty set). If a < 0, then f(x) < 0 for all x ⇒ S = all integers. ∴ The set S must necessarily be either the empty set or the set of all integers.

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