<p>In a triangle ABC, if \(\tan\frac{A}{2}\tan\frac{C}{2} = \frac{1}{3}\) and \(ac = 4\), then the least value of b is:</p><p>(notation have their usual meaning)</p>
Step-by-Step Solution
Key Concept: Use the half-angle tangent formula relation with sides of a triangle, combined with the constraint ac = 4, to express b in terms of a single variable and minimize it using calculus or AM-GM inequality.
<p><strong>Step 1:</strong> Use the half-angle tangent formula for triangles. We know that:</p><p>tan(A/2)tan(C/2) = (s-b)/s, where s = (a+b+c)/2</p><p><strong>Step 2:</strong> Substitute the given condition tan(A/2)tan(C/2) = 1/3:</p><p>(s-b)/s = 1/3</p><p>⟹ 3(s-b) = s</p><p>⟹ 3s - 3b = s</p><p>⟹ 2s = 3b</p><p>⟹ a + b + c = 3b</p><p>⟹ a + c = 2b</p><p><strong>Step 3:</strong> Apply the constraint ac = 4 and the relation a + c = 2b:</p><p>By AM-GM inequality: (a+c)/2 ≥ √(ac)</p><p>⟹ 2b/2 ≥ √4</p><p>⟹ b ≥ 2</p><p><strong>Step 4:</strong> Equality in AM-GM holds when a = c. With ac = 4 and a = c:</p><p>a² = 4 ⟹ a = c = 2</p><p>Then a + c = 4 = 2b ⟹ b = 2</p><p><strong>Step 5:</strong> Verify the triangle inequality: With a = c = 2 and b = 2, we have 2 + 2 > 2 ✓</p><p>∴ The least value of b is 2, and the answer is <strong>b</strong></p>
Correct Answer: b