<p>Let <em>h</em>(<em>x</em>) = 2 − |<em>x</em> − 1| and <em>g</em>(<em>x</em>) = <em>h</em>(|<em>x</em>|) + |<em>h</em>(<em>x</em>)|. Find the number of points where <em>g</em>(<em>x</em>) is non-differentiable.</p>
Step-by-Step Solution
Key Concept: To find non-differentiability points of g(x), we must identify where the absolute value expressions change sign or where composite functions have corner points. These occur where the arguments of absolute values equal zero, and we must check left and right derivatives at each critical point.
<p><strong>Step 1: Find h(x) and its critical points</strong></p><p>h(x) = 2 − |x − 1|</p><p>This has a corner point at x = 1.</p><p>For x ≥ 1: h(x) = 2 − (x − 1) = 3 − x</p><p>For x < 1: h(x) = 2 − (1 − x) = 1 + x</p><p></p><p><strong>Step 2: Find where h(x) = 0</strong></p><p>3 − x = 0 ⟹ x = 3</p><p>1 + x = 0 ⟹ x = −1</p><p>So h(x) changes sign at x = −1 and x = 3.</p><p></p><p><strong>Step 3: Analyze h(|x|)</strong></p><p>h(|x|) = 2 − ||x| − 1|</p><p>This is non-differentiable where |x| − 1 changes sign, i.e., at |x| = 1, giving x = ±1.</p><p></p><p><strong>Step 4: Analyze |h(x)|</strong></p><p>|h(x)| is non-differentiable where h(x) = 0, i.e., at x = −1 and x = 3.</p><p></p><p><strong>Step 5: Find g(x) = h(|x|) + |h(x)|</strong></p><p>Potential non-differentiability points: x = −1, x = 1, and x = 3</p><p></p><p><strong>Step 6: Check each point</strong></p><p><strong>At x = −1:</strong> h(−1) = 0, so |h(−1)| is non-differentiable here. Also, h(|−1|) = h(1) is non-differentiable at x = 1 in the composition, but we evaluate at x = −1. The term |h(x)| has a corner at x = −1. ✓ Non-differentiable</p><p></p><p><strong>At x = 1:</strong> h(|1|) = h(1) involves |x| − 1 = 0, creating a corner. The term h(|x|) is non-differentiable. Also h(1) = 2 − 0 = 2 ≠ 0, so |h(1)| is smooth here. ✓ Non-differentiable</p><p></p><p><strong>At x = 3:</strong> h(3) = 0, so |h(3)| changes from smooth to smooth but has a corner. Also h(|3|) = h(3) is smooth (derivative exists). The term |h(x)| is non-differentiable. ✓ Non-differentiable</p><p></p><p><strong>∴ Answer: 3</strong></p>
Correct Answer: 3