<p><strong>976.</strong> Find the least value of the function \(y = x^2 e^{-x} + 4 - \sqrt{4 - x^2}\).</p>
Step-by-Step Solution
Key Concept: Find critical points by taking the derivative and setting it to zero, then evaluate the function at critical points and boundary points of the domain [-2, 2] to determine the absolute minimum.
<p><strong>Step 1:</strong> Determine the domain. Since √(4 - x²) requires 4 - x² ≥ 0, we have x ∈ [-2, 2].</p><p><strong>Step 2:</strong> Find dy/dx. Let y = x²e^(-x) + 4 - √(4 - x²).</p><p>dy/dx = 2xe^(-x) - x²e^(-x) + x/√(4 - x²)</p><p>dy/dx = xe^(-x)(2 - x) + x/√(4 - x²)</p><p><strong>Step 3:</strong> Setting dy/dx = 0 gives x[e^(-x)(2 - x) + 1/√(4 - x²)] = 0.</p><p>Either x = 0 or e^(-x)(2 - x) + 1/√(4 - x²) = 0.</p><p><strong>Step 4:</strong> Evaluate at critical point x = 0:</p><p>y(0) = 0·e^0 + 4 - √4 = 0 + 4 - 2 = 2</p><p><strong>Step 5:</strong> Evaluate at boundaries:</p><p>y(-2) = 4e² + 4 - 0 = 4e² + 4 ≈ 34.78 (large)</p><p>y(2) = 4e^(-2) + 4 - 0 = 4e^(-2) + 4 ≈ 4.54</p><p><strong>Step 6:</strong> The second derivative test or comparison shows x = 0 gives a local minimum in the interior, and comparing all critical and boundary values:</p><p>∴ Answer: <strong>2</strong> (at x = 0)</p>
Correct Answer: 2