<p>If \(x, y \in R^+\) satisfying \(x + y = 3\), then the maximum value of \(x^2 y\) is ______.</p>
Step-by-Step Solution
Key Concept: For a fixed sum constraint (x + y = 3), use AM-GM inequality or calculus to find that the product x²y is maximized when the weighted contributions are balanced: specifically when 2x = y, which follows from setting the derivative equal to zero.
<p><strong>Step 1:</strong> Use substitution. Given x + y = 3, substitute y = 3 - x where x ∈ (0, 3).</p><p><strong>Step 2:</strong> Rewrite as f(x) = x²(3 - x) = 3x² - x³.</p><p><strong>Step 3:</strong> Find the critical point: f'(x) = 6x - 3x² = 3x(2 - x) = 0.</p><p>This gives x = 0 (boundary) or x = 2 (critical point).</p><p><strong>Step 4:</strong> When x = 2, we have y = 3 - 2 = 1.</p><p><strong>Step 5:</strong> Maximum value = x²y = 2² · 1 = 4.</p><p><strong>Verification:</strong> f''(x) = 6 - 6x; f''(2) = -6 < 0 ✓ (confirms maximum).</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4