<p>Let \ f(x) = x^3 - 3x. The number of solutions of \ f(f(x)) = 0 \ is:</p>
Step-by-Step Solution
Key Concept: Find solutions by first solving f(x) = 0 to get critical values, then solve f(x) = each critical value. The number of real solutions depends on how many times the cubic intersects each horizontal line.
<p><strong>Step 1:</strong> Solve f(x) = 0 where f(x) = x³ - 3x</p><p>x³ - 3x = 0 → x(x² - 3) = 0 → x = 0, √3, -√3</p><p><strong>Step 2:</strong> For f(f(x)) = 0, we need f(x) ∈ {0, √3, -√3}</p><p><strong>Step 3:</strong> Solve f(x) = 0: gives x = 0, ±√3 (3 solutions)</p><p><strong>Step 4:</strong> Solve f(x) = √3: x³ - 3x - √3 = 0</p><p>Since f'(x) = 3x² - 3 = 3(x² - 1), critical points are at x = ±1</p><p>f(-1) = -1 + 3 = 2 (local max), f(1) = 1 - 3 = -2 (local min)</p><p>Since √3 ≈ 1.73 lies between -2 and 2, and the cubic goes from -∞ to +∞, this equation has 3 real solutions</p><p><strong>Step 5:</strong> Solve f(x) = -√3: x³ - 3x + √3 = 0</p><p>By similar analysis (checking that -√3 ≈ -1.73 lies between local min and max), this also has 3 real solutions</p><p><strong>Step 6:</strong> Total solutions = 3 + 3 + 3 = 9</p><p>∴ Answer: D</p>
Correct Answer: D