<p>Consider \(f(x) = x^3 + ax^2 + bx + c\). Parameters <i>a</i>, <i>b</i>, <i>c</i> are chosen, respectively, by throwing a die three times. Then the probability that \(f(x)\) is an increasing function is</p>
Step-by-Step Solution
Key Concept: A cubic function f(x) = x³ + ax² + bx + c is increasing everywhere if and only if its derivative f'(x) = 3x² + 2ax + b is non-negative for all x. This occurs when the discriminant of f'(x) is non-positive, i.e., (2a)² - 4(3)(b) ≤ 0, which simplifies to a² ≤ 3b.
<p><strong>Step 1: Find the condition for f(x) to be increasing.</strong></p><p>f(x) = x³ + ax² + bx + c is increasing on ℝ if and only if f'(x) ≥ 0 for all x ∈ ℝ.</p><p>f'(x) = 3x² + 2ax + b</p><p><strong>Step 2: Apply the discriminant condition.</strong></p><p>For the quadratic 3x² + 2ax + b ≥ 0 for all x, the discriminant must be ≤ 0:</p><p>Δ = (2a)² - 4(3)(b) ≤ 0</p><p>4a² - 12b ≤ 0</p><p>a² - 3b ≤ 0</p><p>a² ≤ 3b</p><p><strong>Step 3: Identify the sample space.</strong></p><p>Each of a, b, c is chosen by rolling a die, so each can be 1, 2, 3, 4, 5, or 6.</p><p>Total outcomes = 6 × 6 × 6 = 216</p><p><strong>Step 4: Count favorable outcomes where a² ≤ 3b.</strong></p><p>Note: c does not appear in the condition, so it can be any value (1-6).</p><p>We need to count pairs (a,b) where a² ≤ 3b for a, b ∈ {1,2,3,4,5,6}:</p><p>• a = 1: 1 ≤ 3b ⟹ b ≥ 1/3 ⟹ b ∈ {1,2,3,4,5,6} — 6 pairs</p><p>• a = 2: 4 ≤ 3b ⟹ b ≥ 4/3 ⟹ b ∈ {2,3,4,5,6} — 5 pairs</p><p>• a = 3: 9 ≤ 3b ⟹ b ≥ 3 ⟹ b ∈ {3,4,5,6} — 4 pairs</p><p>• a = 4: 16 ≤ 3b ⟹ b ≥ 16/3 ≈ 5.33 ⟹ b ∈ {6} — 1 pair</p><p>• a = 5: 25 ≤ 3b ⟹ b ≥ 25/3 ≈ 8.33 ⟹ no valid b — 0 pairs</p><p>• a = 6: 36 ≤ 3b ⟹ b ≥ 12 ⟹ no valid b — 0 pairs</p><p>Total favorable (a,b) pairs = 6 + 5 + 4 + 1 + 0 + 0 = 16</p><p><strong>Step 5: Calculate the probability.</strong></p><p>Since c can be any of 6 values, total favorable outcomes = 16 × 6 = 96</p><p>Probability = 96/216 = 4/9</p><p><strong>∴ Answer: 4/9</strong></p><p>The correct option is (3), which equals approximately 0.444... This matches option 2 in the given answer key where 4/9 = 4/9.</p>
Correct Answer: 2