<p>The minimum value of \(\dfrac{x^4 + y^4 + z^2}{xyz}\) for positive real numbers \(x, y, z\) is</p>
Step-by-Step Solution
Key Concept: Use AM-GM inequality strategically by breaking x⁴ + y⁴ + z² into optimal parts that create a symmetric expression when divided by xyz. The key is recognizing that equality in AM-GM occurs when all terms are equal, which constrains the relationship between x, y, and z.
<p><strong>Step 1:</strong> For positive reals, apply AM-GM inequality to the numerator by breaking it strategically:</p><p>x⁴ + y⁴ + z² = x⁴ + y⁴ + z²</p><p><strong>Step 2:</strong> Use AM-GM on appropriate grouping. Break as: x⁴ + x⁴ + y⁴ + y⁴ + z² ≥ 5∜(x⁸·y⁸·z²) when divided properly, but instead apply directly:</p><p>By AM-GM: (x⁴ + y⁴ + z²)/3 ≥ ∛(x⁴y⁴z²)</p><p>So x⁴ + y⁴ + z² ≥ 3∛(x⁴y⁴z²) = 3(xyz)^(4/3)</p><p><strong>Step 3:</strong> Therefore: (x⁴ + y⁴ + z²)/(xyz) ≥ 3(xyz)^(4/3)/(xyz) = 3(xyz)^(1/3)</p><p><strong>Step 4:</strong> For further minimization, set x = y = z = t. Then:</p><p>(t⁴ + t⁴ + t²)/(t³) = (2t⁴ + t²)/(t³) = 2t + t⁻¹</p><p><strong>Step 5:</strong> Minimize f(t) = 2t + 1/t using calculus: f'(t) = 2 - 1/t² = 0 gives t² = 1/2, so t = 1/√2</p><p><strong>Step 6:</strong> f(1/√2) = 2(1/√2) + √2 = √2 + √2 = 2√2</p><p>∴ Answer: <strong>2√2</strong></p>
Correct Answer: C