<p>A value of \(C\) for which the conclusion of Mean Value Theorem holds for the function \(f(x) = \log_e x\) on the interval [1, 3] is</p>
Step-by-Step Solution
Key Concept: The Mean Value Theorem states that for a continuous and differentiable function on [a,b], there exists c ∈ (a,b) such that f'(c) = [f(b) - f(a)]/(b - a). Apply this directly to find c for f(x) = log_e(x) on [1,3].
<p><strong>Step 1:</strong> Verify conditions: f(x) = ln(x) is continuous on [1,3] and differentiable on (1,3). ✓</p><p><strong>Step 2:</strong> Calculate f'(x) = 1/x</p><p><strong>Step 3:</strong> Apply MVT: f'(c) = [f(3) - f(1)]/(3 - 1)</p><p>1/c = [ln(3) - ln(1)]/(3 - 1) = ln(3)/2</p><p><strong>Step 4:</strong> Solve for c:</p><p>1/c = ln(3)/2</p><p>c = 2/ln(3)</p><p><strong>Step 5:</strong> Verify: c = 2/ln(3) ≈ 2/1.0986 ≈ 1.82, which lies in (1,3) ✓</p><p>∴ Answer: c = 2/ln(3) or equivalently c = 2/log_e(3)</p>
Correct Answer: A