Limits, Continuity & Differentiability
Continuity and Differentiability
Grade 12
Question:
<p>Let \(f\) be a function defined by \(y = f(x)\) where \(x = 2t - |t|\) and \(y = t^2 + t|t|\) for \(t \in \mathbb{R}\), then:</p>
<p>\(f(x)\) is both continuous and differentiable at \(x = 0\).</p>
<p>\(f(x)\) is non differentiable at \(x = 0\).</p>
<p>\(f(x)\) is discontinuous at \(x = 0\).</p>
<p>\(f(x)\) is neither continuous nor differentiable at \(x = 0\).</p>
Step-by-Step Solution
Key Concept: Analyze the parametric function by splitting into cases based on the sign of t, then eliminate the parameter to find the explicit form y = f(x) and determine its properties (continuity, differentiability).
<p><strong>Step 1:</strong> Split into cases based on sign of t.</p><p>For <strong>t ≥ 0:</strong> |t| = t, so x = 2t - t = t and y = t² + t·t = 2t²</p><p>This gives y = 2x² for x ≥ 0</p><p><strong>Step 2:</strong> For <strong>t < 0:</strong> |t| = -t, so x = 2t - (-t) = 3t and y = t² + t·(-t) = 0</p><p>This gives y = 0 for x < 0 (since t < 0 implies x = 3t < 0)</p><p><strong>Step 3:</strong> The function is: f(x) = {0, x < 0; 2x², x ≥ 0}</p><p><strong>Step 4:</strong> Check continuity at x = 0: lim(x→0⁻)f(x) = 0, lim(x→0⁺)f(x) = 0, f(0) = 0. ✓ Continuous everywhere.</p><p><strong>Step 5:</strong> Check differentiability at x = 0: Left derivative = 0, Right derivative = 0. ✓ Differentiable everywhere.</p><p>∴ f is continuous and differentiable on ℝ</p>
Correct Answer: A