<p>Given function \(f(x) = x^3 + x^2 f'(1) + xf''(2) + f'''(3)\). Then \(f(2)\) equals:</p>
Step-by-Step Solution
Key Concept: The function f(x) contains its own derivatives as coefficients, so differentiate f(x) to get equations relating f'(1), f''(2), and f'''(3), then solve this self-referential system by comparing coefficients.
<p><strong>Step 1:</strong> Let f(x) = x³ + x²f'(1) + xf''(2) + f'''(3), where f'(1), f''(2), f'''(3) are constants to determine.</p><p><strong>Step 2:</strong> Differentiate: f'(x) = 3x² + 2xf'(1) + f''(2)</p><p><strong>Step 3:</strong> Differentiate again: f''(x) = 6x + 2f'(1)</p><p><strong>Step 4:</strong> Differentiate once more: f'''(x) = 6</p><p><strong>Step 5:</strong> Since f'''(x) = 6 (constant), we have f'''(3) = 6</p><p><strong>Step 6:</strong> From f''(x) = 6x + 2f'(1), at x = 2: f''(2) = 12 + 2f'(1)</p><p><strong>Step 7:</strong> From f'(x) = 3x² + 2xf'(1) + f''(2), at x = 1: f'(1) = 3 + 2f'(1) + f''(2)</p><p>This gives: -f'(1) = 3 + f''(2), so f'(1) = -3 - f''(2)</p><p><strong>Step 8:</strong> Substitute into f''(2) = 12 + 2f'(1):<br/>f''(2) = 12 + 2(-3 - f''(2)) = 12 - 6 - 2f''(2)<br/>3f''(2) = 6, so f''(2) = 2</p><p><strong>Step 9:</strong> Therefore f'(1) = -3 - 2 = -5</p><p><strong>Step 10:</strong> Now f(x) = x³ - 5x² + 2x + 6</p><p><strong>Step 11:</strong> f(2) = 8 - 5(4) + 2(2) + 6 = 8 - 20 + 4 + 6 = -2</p><p>∴ Answer: B</p>
Correct Answer: B