Question:

<div class="solution"><p><strong>Key Idea:</strong> Show min{sin x, |x|} = sin x for all x.</p><p><strong>Step 1:</strong> For x≥0: sin x ≤ x = |x|; for x<0: sin x ≤ 0 < |x|. So min = sin x always.</p><p><strong>Step 2:</strong> <span class="math-inline">$f(x)=\sin x + \{x/\pi\}$</span></p><p><strong>Step 3:</strong> Test π: <span class="math-inline">$f(x+\pi)=-\sin x+\{x/\pi\}\ne f(x)$</span>. So π not a period.</p><p><strong>Step 4:</strong> Test 2π: <span class="math-inline">$f(x+2\pi)=\sin x+\{x/\pi+2\}=f(x)$</span> ✓</p><p><strong>Answer: 2π</strong></p><div class="trap-box"><strong>Trap:</strong> {x/π} alone has period π, but the combined function does not. Always test the full expression.</div><div class="key-concept"><strong>Key Concept:</strong> Period of sum — LCM approach, but always verify directly</div></div>