<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: Since f(x) is a polynomial and f'(x) + f(x) = x, recognize that f(x) must be a linear polynomial (degree 1). Equate coefficients on both sides after substituting f(x) = ax + b.
<p><strong>Step 1:</strong> Since f(x) is a polynomial and f'(x) + f(x) = x, the degree of f(x) must be 1 (higher degrees would create inconsistency). Let f(x) = ax + b.</p><p><strong>Step 2:</strong> Then f'(x) = a. Substitute into the given equation:</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> f(4) = 4 - 1 = 3</p><p>∴ Answer: C</p>
Correct Answer: C