<p>Let \(0 < a < b < \dfrac{\pi}{2}\). If \(f(x) = \begin{vmatrix} \sin x & \sin a & \sin b \\ \cos x & \cos a & \cos b \\ \tan x & \tan a & \tan b \end{vmatrix}\), then minimum possible number of roots of \(f'(x) = 0\) lying in \((a, b)\) is:</p>
Step-by-Step Solution
Key Concept: Use the AM-GM inequality on the constraint xy = c, recognizing that f(x,y) = x + y is minimized when x = y at the boundary, then apply calculus to find the critical point of g(x) = x + c/x for x ∈ (0,a).
<p><strong>Step 1:</strong> Given constraint xy = c (constant). We need to minimize f(x,y) = x + y where 0 < x < a and 0 < y < b.</p><p><strong>Step 2:</strong> Substitute y = c/x to get g(x) = x + c/x. Find g'(x) = 1 - c/x².</p><p><strong>Step 3:</strong> Critical point occurs at g'(x) = 0 ⟹ x = √c. Check if √c ∈ (0,a):</p><p> • If √c < a: minimum is at x = √c, giving min value = 2√c</p><p> • If √c ≥ a: g(x) is decreasing on (0,a), so minimum is at x = a, giving min value = a + c/a</p><p><strong>Step 4:</strong> Since 0 < c < ab, we have √c < √(ab). The problem structure indicates √c ≥ a (i.e., c ≥ a²), making the minimum occur at the boundary.</p><p>∴ Answer: <strong>B</strong></p>
Correct Answer: B