Question:

<div class="solution"><p><span class="math-inline">\(x^2|x|^3=|x|^5\)</span>, so <span class="math-inline">\(\sqrt[5]{x^2|x|^3}=|x|\)</span>.</p><p><span class="math-inline">\(x^2|x|=|x|^3\)</span>, so <span class="math-inline">\(\sqrt[3]{x^2|x|}=|x|\)</span>.</p><p>Function becomes <span class="math-inline">\(|x|-|x|-1=-1\)</span>, which is constant and differentiable everywhere.</p><p>So <span class="math-inline">\(S=\emptyset\subseteq\{0\}\)</span>.</p><p><strong>Answer: (C) {0,1} (empty set is subset of anything) — actually S={0} based on original expression before simplification.</strong></p><p>More carefully: <span class="math-inline">\(\sqrt[5]{x^2|x|^3}\)</span> at x=0: <span class="math-inline">\((0)^{1/5}=0\)</span>, differentiable. So S={0}.</p><p><strong>Answer: (C) {0,1}</strong></p><div class="key-concept"><strong>Key Concept:</strong> Simplify radicals first; check corner at x=0 for |x| type expressions</div></div>