<p>Let <em>f</em> : (−1, 1) → <em>R</em> be a function defined by \( f(x) = \max\left\{-|x|, -\sqrt{1-x^2}\right\} \). If <em>K</em> be the set of all points at which <em>f</em> is not differentiable, then <em>K</em> has exactly:</p>
Step-by-Step Solution
Key Concept: The function f(x) is the pointwise maximum of two continuous functions: -|x| and -√(1-x²). Non-differentiability occurs where these two curves intersect or where either individual function is non-differentiable. We must find intersection points and check corner points.
<p><strong>Step 1:</strong> Identify the two component functions:</p><ul><li>g(x) = -|x|: non-differentiable at x = 0</li><li>h(x) = -√(1-x²): differentiable on (-1,1)</li></ul><p><strong>Step 2:</strong> Determine which function dominates in each region by finding intersections. Set -|x| = -√(1-x²):</p><p>|x| = √(1-x²) ⟹ x² = 1-x² ⟹ 2x² = 1 ⟹ x = ±1/√2</p><p><strong>Step 3:</strong> Analyze f(x) in regions:</p><ul><li>For x ∈ (-1, -1/√2): -√(1-x²) < -|x|, so f(x) = -|x|</li><li>For x ∈ (-1/√2, 0): -|x| > -√(1-x²), so f(x) = -√(1-x²)</li><li>For x ∈ (0, 1/√2): -|x| > -√(1-x²), so f(x) = -√(1-x²)</li><li>For x ∈ (1/√2, 1): -√(1-x²) < -|x|, so f(x) = -|x|</li></ul><p><strong>Step 4:</strong> Check differentiability at critical points:</p><ul><li>At x = -1/√2 and x = 1/√2: The two functions meet with different slopes (one has slope from -|x|, other from h), causing non-differentiability ✓</li><li>At x = 0: Left of 0, f(x) = -√(1-x²) with f'(0⁻) = 0; Right of 0, f(x) = -√(1-x²) with f'(0⁺) = 0. The function is smooth here ✗</li></ul><p><strong>Step 5:</strong> K = {-1/√2, 1/√2} contains exactly <strong>2 points</strong></p><p>∴ Answer: C (2 points)</p>
Correct Answer: C