<p><strong>283.</strong> If inequality \( \left(\dfrac{1}{x}\right)^{\lambda/x} \leq \dfrac{1}{9} \) has positive integer solution, then the minimum value of \( \lambda \) (using \( \ln 9 = 2.197 \)) is:</p>
Step-by-Step Solution
Key Concept: Rewrite the inequality as x^(-λ/x) ≤ 9^(-1), then take logarithms to get -λln(x)/x ≤ -ln(9), which simplifies to λ ≥ x·ln(9)/ln(x). Find the minimum value of λ by determining which positive integer x minimizes the right side.
<p><strong>Step 1:</strong> Rewrite the inequality (1/x)^(λ/x) ≤ 1/9</p><p>Taking natural logarithm on both sides: (λ/x)·ln(1/x) ≤ ln(1/9)</p><p><strong>Step 2:</strong> Simplify: (λ/x)·(-ln x) ≤ -ln 9</p><p>-λ·ln(x)/x ≤ -ln 9</p><p>λ·ln(x)/x ≥ ln 9</p><p><strong>Step 3:</strong> Rearrange: λ ≥ x·ln(9)/ln(x)</p><p><strong>Step 4:</strong> For positive integer solutions, test values:</p><p>• x = 1: ln(1) = 0, undefined (division by 0)</p><p>• x = 2: λ ≥ 2·ln(9)/ln(2) = 2·2.197/0.693 ≈ 6.34</p><p>• x = 3: λ ≥ 3·ln(9)/ln(3) = 3·2.197/1.099 ≈ 6.00</p><p>• x = 4: λ ≥ 4·ln(9)/ln(4) = 4·2.197/1.386 ≈ 6.34</p><p>• x = 9: λ ≥ 9·ln(9)/ln(9) = 9·2.197/2.197 = 9</p><p><strong>Step 5:</strong> The minimum value of x·ln(9)/ln(x) occurs at x = 3, giving λ_min ≈ 6</p><p>∴ Answer: B</p>
Correct Answer: B