<p>Let <i>M</i> be the greatest and <i>m</i> be the least value of \(\sqrt{\sin^{-1} x} + \sqrt{\cos^{-1} x}\), then find the value of \((M/m)^4\).</p>
Step-by-Step Solution
Key Concept: Use substitution sin⁻¹x + cos⁻¹x = π/2 to convert the sum of square roots into a single variable optimization problem. Then apply AM-GM or calculus to find extrema.
<p><strong>Step 1:</strong> Use the identity sin⁻¹x + cos⁻¹x = π/2 for x ∈ [-1, 1].</p><p>Let sin⁻¹x = θ, then cos⁻¹x = π/2 - θ where θ ∈ [-π/2, π/2].</p><p><strong>Step 2:</strong> The function becomes f(θ) = √θ + √(π/2 - θ) for θ ∈ [0, π/2] (domain restriction where both terms are real).</p><p><strong>Step 3:</strong> Find critical points: f'(θ) = 1/(2√θ) - 1/(2√(π/2 - θ)) = 0</p><p>This gives √(π/2 - θ) = √θ, so θ = π/4.</p><p><strong>Step 4:</strong> Evaluate at critical point and boundaries:</p><p>• At θ = π/4: f(π/4) = √(π/4) + √(π/4) = 2√(π/4) = √π</p><p>• At θ = 0: f(0) = 0 + √(π/2) = √(π/2)</p><p>• At θ = π/2: f(π/2) = √(π/2) + 0 = √(π/2)</p><p><strong>Step 5:</strong> Maximum M = √π, Minimum m = √(π/2)</p><p><strong>Step 6:</strong> (M/m)⁴ = [√π / √(π/2)]⁴ = [√2]⁴ = (√2)⁴ = 4</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4