Applications of Derivatives
Maxima and Minima
Grade 12
Question:
<p>We have <br/>\[u^2 = a^2 + b^2 + 2\sqrt{(a^4+b^4)\sin^2\theta\cos^2\theta + a^2b^2(\sin^4\theta + \cos^4\theta)}\]<br/> The minimum value of \(u^2\) is:</p>
<p>(1) \(a^2 + b^2 + 2a^2b^2\)</p>
<p>(2) \((a+b)^2\)</p>
<p>(3) \(a^2 + b^2\)</p>
<p>(4) \(a^2 + b^2 + 2ab\)</p>
Step-by-Step Solution
Key Concept: Recognize that the expression under the square root can be rewritten by analyzing the term (a⁴+b⁴)sin²θcos²θ + a²b²(sin⁴θ+cos⁴θ) as a quadratic in sin²θcos²θ. The minimum occurs when we optimize over θ using calculus or algebraic simplification.
<p><strong>Step 1:</strong> Simplify sin⁴θ + cos⁴θ. We know sin⁴θ + cos⁴θ = (sin²θ + cos²θ)² - 2sin²θcos²θ = 1 - 2sin²θcos²θ</p><p><strong>Step 2:</strong> Let t = sin²θcos²θ where 0 ≤ t ≤ 1/4. Substitute into the radical:</p><p>Radical = (a⁴+b⁴)t + a²b²(1 - 2t) = (a⁴+b⁴)t + a²b² - 2a²b²t = a²b² + t(a⁴+b⁴-2a²b²) = a²b² + t(a²-b²)²</p><p><strong>Step 3:</strong> Since t ≥ 0, the minimum value of the radical occurs at t = 0:</p><p>Minimum of radical = a²b² = |ab|</p><p><strong>Step 4:</strong> Therefore, u² = a² + b² + 2|ab| = (|a| + |b|)²</p><p>Minimum value of u² = <strong>(a+b)²</strong> or equivalently <strong>a² + b² + 2ab</strong></p><p>∴ Answer: D</p>
Correct Answer: D