Limits, Continuity & Differentiability
Differentiability
Grade 12
Question:
<p>The set of points where \(f(x) = \frac{x}{4 + |x|}\) is differentiable is</p>
<p>(a) \((-\infty, +\infty)\)</p>
<p>(b) \((0, +\infty)\)</p>
<p>(c) \((-\infty, 0) \cup (0, +\infty)\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: A function involving absolute value is non-differentiable at points where the absolute value expression changes sign. Check differentiability at x=0 by examining left and right derivatives using the definition.
<p><strong>Step 1:</strong> Identify where the absolute value changes sign. The function <em>f(x) = x/(4+|x|)</em> has |x| which changes behavior at <em>x = 0</em>.</p><p><strong>Step 2:</strong> For <em>x > 0</em>: <em>f(x) = x/(4+x)</em>, so <em>f'(x) = (4+x-x)/(4+x)² = 4/(4+x)²</em>. Right derivative at 0: <em>f'<sub>R</sub>(0) = 4/16 = 1/4</em>.</p><p><strong>Step 3:</strong> For <em>x < 0</em>: <em>f(x) = x/(4-x)</em>, so <em>f'(x) = (4-x+x)/(4-x)² = 4/(4-x)²</em>. Left derivative at 0: <em>f'<sub>L</sub>(0) = 4/16 = 1/4</em>.</p><p><strong>Step 4:</strong> Since <em>f'<sub>L</sub>(0) = f'<sub>R</sub>(0) = 1/4</em>, the function is differentiable at <em>x = 0</em>. For all other points, the function is smooth since no absolute value sign changes occur.</p><p>∴ Answer: <em>f</em> is differentiable at all points in <em>ℝ</em> (or <em>(-∞, ∞)</em>)</p>
Correct Answer: A