Question:

<div class="solution"><p><strong>sgn(x):</strong> = -1 for x<0, 0 for x=0, 1 for x>0.</p><p><strong>For x>0:</strong> <span class="math-inline">\(f(x)=|x-1|\)</span> — non-differentiable at x=1.</p><p><strong>For x<0:</strong> <span class="math-inline">\(f(x)=|x+1|\)</span> — non-differentiable at x=-1.</p><p><strong>At x=0:</strong> <span class="math-inline">\(f(0)=|0-0|=0\)</span>. LHD: <span class="math-inline">\(f(x)=|x+1|\to 1\)</span> near 0⁻, derivative = sign(0+1)·1=1. RHD: <span class="math-inline">\(f(x)=|x-1|\to 1\)</span> near 0⁺, derivative = sign(0-1)·1=-1. LHD≠RHD → non-differentiable at x=0.</p><p>Total: <strong>3 points</strong> (x=-1, 0, 1).</p><p><strong>Answer: (D) 3</strong></p><div class="trap-box"><strong>Trap:</strong> Missing x=0 as a non-differentiable point.</div><div class="key-concept"><strong>Key Concept:</strong> Check corners from both |expression| AND from sgn discontinuity</div></div>