<p>Let \(f(x)\) be a polynomial function satisfying \(f'(x) + f(x) = x\). Then the value of \(f(4)\) is equal to:</p>
Step-by-Step Solution
Key Concept: For a polynomial f(x) satisfying f'(x) + f(x) = x, assume f(x) has the same degree as the RHS. Comparing coefficients of like powers on both sides reveals the polynomial structure, then use initial/boundary conditions to find constants.
<p><strong>Step 1:</strong> Since f(x) is a polynomial and the RHS is degree 1, assume f(x) = ax + b (degree 1 polynomial).</p><p><strong>Step 2:</strong> Then f'(x) = a. Substituting into f'(x) + f(x) = x:</p><p>a + (ax + b) = x</p><p>ax + (a + b) = x</p><p><strong>Step 3:</strong> Comparing coefficients:</p><p>• Coefficient of x: a = 1</p><p>• Constant term: a + b = 0 → b = -1</p><p><strong>Step 4:</strong> Therefore f(x) = x - 1</p><p><strong>Step 5:</strong> Verify: f'(x) = 1, and f'(x) + f(x) = 1 + (x - 1) = x ✓</p><p><strong>Step 6:</strong> f(4) = 4 - 1 = 3</p><p>∴ Answer: C (assuming C = 3)</p>
Correct Answer: C