<p>If three positive real numbers <i>a</i>, <i>b</i>, <i>c</i> are in A.P. such that \(abc = 4\), then the minimum value of <i>b</i> is</p>
Step-by-Step Solution
Key Concept: Since a, b, c are in A.P., we have b = (a+c)/2. Using the AM-GM inequality on a and c, combined with the constraint abc = 4, we can find the minimum value of b.
<p><strong>Step 1:</strong> Since a, b, c are in A.P., we have:</p><p>2b = a + c ... (1)</p><p><strong>Step 2:</strong> Given constraint: abc = 4 ... (2)</p><p><strong>Step 3:</strong> From (1): a + c = 2b. By AM-GM inequality on positive numbers a and c:</p><p>$$\frac{a+c}{2} \geq \sqrt{ac}$$</p><p>$$b \geq \sqrt{ac}$$</p><p>$$b^2 \geq ac$$ ... (3)</p><p><strong>Step 4:</strong> From constraint (2): ac = 4/b</p><p><strong>Step 5:</strong> Substituting in (3):</p><p>$$b^2 \geq \frac{4}{b}$$</p><p>$$b^3 \geq 4$$</p><p>$$b \geq \sqrt[3]{4}$$</p><p><strong>Step 6:</strong> Equality in AM-GM occurs when a = c. Combined with 2b = a + c, we get a = b = c.</p><p>From abc = 4: b³ = 4, so b = ∛4 = 2^(2/3)</p><p>∴ <strong>Minimum value of b is ∛4 (or 2^(2/3))</strong></p>
Correct Answer: B