Limits, Continuity & Differentiability
Non-differentiable Points
Grade 12
Question:
<p>Let <i>f(x) = ||x| − 1|</i>, then points where <i>f(x)</i> is not differentiable is/are</p>
<p>(a) 0, ±1</p>
<p>(b) ±1</p>
<p>(c) 0</p>
<p>(d) 1</p>
Step-by-Step Solution
Key Concept: Absolute value functions are not differentiable where the argument changes sign; nested absolute values create multiple non-differentiable points
<p>The function <i>f(x) = ||x| − 1|</i> involves nested absolute values. To find non-differentiability points, analyze where the expression inside absolute value changes sign:</p><p>Inner: <i>|x|</i> has a corner at <i>x = 0</i></p><p>Middle: <i>|x| − 1 = 0</i> when <i>x = ±1</i></p><p>Therefore <i>f(x)</i> is not differentiable at <i>x = 0, ±1</i></p>
Correct Answer: a