Limits, Continuity & Differentiability
Piecewise Defined Functions
Grade 12
Question:
<p>If <i>f(x) = min {1, x², x³}</i>, then</p>
<p>(a) <i>f(x)</i> is continuous everywhere</p>
<p>(b) <i>f(x)</i> is differentiable everywhere except at two points</p>
<p>(c) <i>f(x)</i> is not differentiable at one point</p>
<p>(d) <i>f(x)</i> is not differentiable at one point</p>
Step-by-Step Solution
Key Concept: For min/max functions, find where different pieces take over and check differentiability at transition points
<p>Analyze the minimum function:</p><p>For <i>x ≤ −1</i>: <i>f(x) = x³</i> (since <i>x³ ≤ x² ≤ 1</i>)</p><p>For <i>−1 ≤ x ≤ 0</i>: <i>f(x) = x³</i></p><p>For <i>0 ≤ x ≤ 1</i>: <i>f(x) = x²</i> (since <i>x² ≤ 1, x³</i>)</p><p>For <i>x ≥ 1</i>: <i>f(x) = 1</i></p><p>At <i>x = 1</i>: left derivative is <i>2(1) = 2</i>, right derivative is 0. Not differentiable.</p><p>At <i>x = −1</i> and <i>x = 0</i>: both derivatives match (3x²).</p><p>Therefore, <i>f(x)</i> is not differentiable at exactly one point: <i>x = 1</i></p>
Correct Answer: c