<p>Let \(f(x) = ax^2 - bx + c^2\), \(b \neq 0\) and \(f(x) \neq 0\) for all \(x \in \mathbb{R}\). Then</p>
<p>(1) \(a + c^2 < b\)</p>
<p>(2) \(4a + c^2 > 2b\)</p>
<p>(3) \(9a - 3b + c^2 < 0\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: If a quadratic f(x) ≠ 0 for all real x, its discriminant must be negative AND the leading coefficient must have the same sign as the constant term. Here, using discriminant < 0 and analyzing the inequality b² < 4ac² reveals that a and c² must have the same sign relationship, which forces a > 0.
<p><strong>Step 1:</strong> Since f(x) = ax² - bx + c² ≠ 0 for all x ∈ ℝ, the quadratic has no real roots.</p><p><strong>Step 2:</strong> This means discriminant Δ = b² - 4ac² < 0, so b² < 4ac².</p><p><strong>Step 3:</strong> Since b² ≥ 0 and we need b² < 4ac², we require 4ac² > 0. Since c² ≥ 0, we must have a > 0.</p><p><strong>Step 4:</strong> Also, f(0) = c² > 0 (given b ≠ 0 ensures the quadratic is non-trivial, and f(x) ≠ 0 for all x means c ≠ 0).</p><p><strong>Step 5:</strong> From b² < 4ac² with a > 0 and c² > 0, we get a > 0, and similarly b² < 4ac² implies a/c² > b²/(4c⁴), confirming a > 0.</p><p>∴ The correct statements typically include: <strong>a > 0</strong> and statements about the relationship between a, b, and c.</p>
Correct Answer: B