<p><strong>101.</strong> Slope of tangent to the curve \(y = 2e^x \sin\!\left(\dfrac{\pi}{4} - \dfrac{x}{2}\right)\cos\!\left(\dfrac{\pi}{4} - \dfrac{x}{2}\right)\) where \(0 \leq x \leq 2\pi\), is minimum at \(x\) is equal to:</p>
Step-by-Step Solution
Key Concept: Simplify the trigonometric expression using double angle formula (2sin A cos A = sin 2A), then find dy/dx and minimize it by setting d²y/dx² = 0.
<p><strong>Step 1:</strong> Simplify using double angle formula: 2sin(π/4 - x/2)cos(π/4 - x/2) = sin(π/2 - x) = cos x</p><p>So y = 2e^x cos x</p><p><strong>Step 2:</strong> Find dy/dx using product rule: dy/dx = 2e^x cos x + 2e^x(-sin x) = 2e^x(cos x - sin x)</p><p><strong>Step 3:</strong> To minimize slope, find d²y/dx² and set it equal to 0:</p><p>d²y/dx² = 2e^x(cos x - sin x) + 2e^x(-sin x - cos x) = 2e^x(-2sin x) = -4e^x sin x</p><p><strong>Step 4:</strong> Set d²y/dx² = 0: -4e^x sin x = 0</p><p>Since e^x > 0, we need sin x = 0</p><p>For 0 ≤ x ≤ 2π: x = 0, π, 2π</p><p><strong>Step 5:</strong> Check which gives minimum slope (not minimum value of second derivative, but minimum value of dy/dx itself):</p><p>At x = 0: dy/dx = 2(1) = 2</p><p>At x = π: dy/dx = 2e^π(-1 - 0) = -2e^π (minimum)</p><p>At x = 2π: dy/dx = 2e^(2π)(1) = 2e^(2π)</p><p>∴ Answer: B (x = π)</p>
Correct Answer: B