<p>The maximum value of the expression \(\dfrac{x^m y^n}{(1+x^{2m})(1+y^{2n})}\) is:</p>
Step-by-Step Solution
Key Concept: Use AM-GM inequality on the denominator terms to find when the ratio is maximized. The maximum occurs when the denominator is minimized relative to the numerator, which happens at a specific relationship between x and y.
<p><strong>Step 1:</strong> Let f(x,y) = x^m y^n / [(1+x^2m)(1+y^2n)]. We need to find its maximum value.</p><p><strong>Step 2:</strong> By AM-GM inequality: 1 + x^2m ≥ 2√(x^2m) = 2x^m (equality when x^2m = 1, i.e., x = 1)</p><p><strong>Step 3:</strong> Similarly: 1 + y^2n ≥ 2√(y^2n) = 2y^n (equality when y^2n = 1, i.e., y = 1)</p><p><strong>Step 4:</strong> Therefore: f(x,y) ≤ (x^m · y^n)/(2x^m · 2y^n) = 1/4</p><p><strong>Step 5:</strong> Equality holds when x = 1 and y = 1 simultaneously.</p><p><strong>Step 6:</strong> At x = 1, y = 1: f(1,1) = (1)(1)/[(1+1)(1+1)] = 1/4</p><p>∴ Maximum value = <strong>1/4</strong></p>
Correct Answer: B