Limits, Continuity & Differentiability
Continuity and Differentiability
Grade 12
Question:
<p>\(f(x) = x^2 + 3, x \leq 1\)<br>\(= 3x + a, x > 1\)<br>Is \(f(x)\) neither continuous nor differentiable at \(x = 1\)?</p>
<p>(a) True</p>
<p>(b) False</p>
Step-by-Step Solution
Key Concept: A function is continuous at x=1 if left limit = right limit = f(1). For differentiability, the left and right derivatives must also be equal. We need to find the value of 'a' that makes f continuous, then check if the derivatives match.
<p><strong>Step 1: Apply Continuity Condition</strong></p><p>For continuity at x = 1:</p><p>Left limit: lim(x→1⁻) f(x) = 1² + 3 = 4</p><p>f(1) = 1² + 3 = 4</p><p>Right limit: lim(x→1⁺) f(x) = 3(1) + a = 3 + a</p><p>For continuity: 3 + a = 4 ⟹ <strong>a = 1</strong></p><p><strong>Step 2: Check Differentiability at x = 1 (with a = 1)</strong></p><p>Left derivative: f'(1⁻) = d/dx(x² + 3) = 2x|ₓ₌₁ = 2</p><p>Right derivative: f'(1⁺) = d/dx(3x + 1) = 3</p><p><strong>Step 3: Conclusion</strong></p><p>Since f'(1⁻) = 2 ≠ 3 = f'(1⁺), the derivatives do not match.</p><p>Therefore, f(x) is <strong>continuous but NOT differentiable</strong> at x = 1 when a = 1.</p><p>The statement 'neither continuous nor differentiable' is <strong>FALSE</strong>.</p><p>∴ Answer: B</p>
Correct Answer: B