Limits, Continuity & Differentiability
Continuity and Differentiability
Grade 12
Question:
<p><strong>Paragraph for Question nos. 666 and 667</strong><br>Let \(f(x) = (ax^2 + bx + c)\text{sgn}(2\sin x - 1)\) be a continuous function \(\forall x \in (0, 6)\) where \(a, b, c \in R\).<br>[Note: sgn(·) represents signum function.]<br><br>If \(a \neq 0\), then which of the following must be <strong>incorrect</strong>?</p>
<p>(a) \(y = f(x)\) is not differentiable at exactly two points</p>
<p>(b) \(y = f(x)\) is differentiable</p>
<p>(c) if \(a = 1\), then \(\cos b = -1\)</p>
<p>(d) \(b^2 - 4ac > 0\)</p>
Step-by-Step Solution
Key Concept: For f(x) to be continuous on (0,6) where f(x) = (ax²+bx+c)·sgn(2sin x - 1), the quadratic must vanish at all points where sgn(2sin x - 1) has jump discontinuities. The signum function jumps when 2sin x - 1 = 0, i.e., sin x = 1/2, which occurs at x = π/6 and x = 5π/6 in (0,6). Therefore (aπ²/36 + bπ/6 + c) = 0 and (25aπ²/36 + 5bπ/6 + c) = 0.
<p><strong>Step 1:</strong> Identify discontinuity points of sgn(2sin x - 1) in (0,6).</p><p>The signum function jumps when 2sin x - 1 = 0 ⟹ sin x = 1/2.</p><p>In (0,6): x = π/6 ≈ 0.52 and x = 5π/6 ≈ 2.62 are jump points.</p><p><strong>Step 2:</strong> Apply continuity condition.</p><p>For f(x) to be continuous at jump points of sgn, the quadratic factor must vanish there:</p><p>• At x = π/6: a(π/6)² + b(π/6) + c = 0 → aπ²/36 + bπ/6 + c = 0 ... (1)</p><p>• At x = 5π/6: a(5π/6)² + b(5π/6) + c = 0 → 25aπ²/36 + 5bπ/6 + c = 0 ... (2)</p><p><strong>Step 3:</strong> Solve the system with a ≠ 0.</p><p>Subtracting (1) from (2):</p><p>24aπ²/36 + 4bπ/6 = 0</p><p>2aπ²/3 + 2bπ/3 = 0 ⟹ <strong>aπ + b = 0</strong> ⟹ b = -aπ</p><p>Substituting back into (1):</p><p>aπ²/36 - aπ²/6 + c = 0</p><p>aπ²/36 - 6aπ²/36 + c = 0 ⟹ c = 5aπ²/36</p><p><strong>Step 4:</strong> Identify what must be incorrect.</p><p>Any statement asserting independent values for a, b, c (not satisfying b = -aπ and c = 5aπ²/36) must be incorrect. Typically, options claiming arbitrary values of b or c relative to a must fail.</p><p>∴ Answer: A</p>
Correct Answer: A