Question:

<div class="solution"><p><strong>Step 1:</strong> Recognize \(\sin^{-1}\!\sqrt{x/2}+\cos^{-1}\!\sqrt{x/2}=\pi/2\) (complementary pair).</p><p><strong>Step 2:</strong> \(\pi/2+\tan^{-1}y=2\pi/3\implies\tan^{-1}y=\pi/6\implies y=1/\sqrt{3}\).</p><p><strong>Step 3:</strong> Domain: \(0\le x\le 4\). So \(x^2\in[0,16]\).</p><p>Max: \(16+1/3=49/3\) ✓. Min: \(0+1/3=1/3\) ✓.</p><p><strong>Answer: Both (A) and (B)</strong></p><div class="trap-box"><strong>Trap:</strong> The first two terms don't vary independently -- they collapse to a constant via the complementary identity.<div class="key-concept"><strong>Key Concept:</strong> Complementary sin⁻^1+cos⁻^1=\pi/2 collapses multi-variable equations