Quadratic Equations
Roots and Coefficients
Grade 11
Question:
<p>If <strong>f</strong>(<strong>x</strong>) = <strong>ax</strong><sup>2</sup> − <strong>bx</strong> + <strong>c</strong> has two distinct roots α and β, and f(0) and f(1) are of the same sign, with α(1 − α) ≤ 1/4 for all α ∈ (0,1), find the least value of <strong>a</strong>.</p>
Step-by-Step Solution
Key Concept: Use the condition that f(0) and f(1) have the same sign to establish bounds on a, combined with the constraint that α(1 − α) ≤ 1/4 to find the minimum integer value.
<p><strong>Step 1:</strong> Since f(0) and f(1) are of the same sign: c(a − b + c) > 0 ⟹ c(a − b + c) ≥ 1</p><p><strong>Step 2:</strong> Therefore a²αβ(1 − α)(1 − β) ≥ 1</p><p><strong>Step 3:</strong> Since α(1 − α) = 1/4 − (α − 1/2)² ≤ 1/4, we get a²αβ(1 − α)(1 − β) < a²/16</p><p><strong>Step 4:</strong> Thus a²/16 > 1 ⟹ a > 4. Since a ∈ ℤ, we have a ≥ 5</p><p><strong>Step 5:</strong> Also from b² − 4ac ≥ 0, we get b² ≥ 4ac ≥ 20 ⟹ b ≥ 5</p><p><strong>Step 6:</strong> With a ≥ 5, b ≥ 5, we get c ≥ 1, so abc ≥ 25</p><p>∴ <strong>Least value of a is 5</strong></p>
Correct Answer: 5