<p>Let <em>m</em> and <em>n</em> be positive real numbers such that <em>m</em> + <em>n</em> = 3. If \(\frac{m}{s} = \sin^2\theta\) and \(\frac{n}{t} = \cos^2\theta\), then the minimum value of \(s + t\) is:</p>
Step-by-Step Solution
Key Concept: Use the constraint m + n = 3 with sin²θ + cos²θ = 1 to express s + t in terms of a single variable, then apply AM-GM or calculus to find the minimum.
<p><strong>Step 1:</strong> Write the given conditions:<br>m + n = 3, where m, n > 0<br>m/s = sin²θ and n/t = cos²θ<br>Therefore: s = m/sin²θ and t = n/cos²θ</p><p><strong>Step 2:</strong> Use the fundamental identity sin²θ + cos²θ = 1:<br>s + t = m/sin²θ + n/cos²θ<br>Let sin²θ = x, so cos²θ = 1 - x, where 0 < x < 1<br>s + t = m/x + n/(1-x) where m + n = 3</p><p><strong>Step 3:</strong> By Cauchy-Schwarz inequality (or weighted AM-HM):<br>(m/x + n/(1-x))(x + (1-x)) ≥ (√m + √n)²<br>(m/x + n/(1-x)) · 1 ≥ (√m + √n)²</p><p><strong>Step 4:</strong> Alternatively, use calculus. For fixed m, n with m + n = 3:<br>f(x) = m/x + n/(1-x)<br>f'(x) = -m/x² + n/(1-x)² = 0<br>This gives: m(1-x)² = nx²</p><p><strong>Step 5:</strong> By symmetry and optimization, minimum occurs when m = n = 3/2 and sin²θ = cos²θ = 1/2:<br>s + t = (3/2)/(1/2) + (3/2)/(1/2) = 3 + 3 = 6</p><p><strong>Step 6:</strong> Verify using Cauchy-Schwarz directly:<br>(m/sin²θ + n/cos²θ)(sin²θ + cos²θ) ≥ (√m + √n)²<br>s + t ≥ (√m + √n)² ≥ (2√(mn)) when m = n = 3/2<br>Minimum: s + t = (√(3/2) + √(3/2))² = (√6)² = 6</p><p>∴ Answer: A (The minimum value is <strong>6</strong>)</p>
Correct Answer: A