Limits, Continuity & Differentiability
Continuity and Differentiability of Composite Functions
Grade 12
Question:
<p>If <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) = max(|3 + <span style="font-style: italic;">x</span>|, 3 + <span style="font-style: italic;">x</span>³), then which of the following is/are correct?</p>
<p>(a) <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is continuous for all <span style="font-style: italic;">x</span> ∈ ℝ</p>
<p>(b) <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is differentiable for all <span style="font-style: italic;">x</span> ∈ ℝ</p>
<p>(c) <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is non-differentiable at three points only</p>
<p>(d) <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is non-differentiable at four points only</p>
Step-by-Step Solution
Key Concept: The maximum of two functions is continuous wherever both are continuous. Non-differentiability occurs at points where the two constituent functions intersect and have different derivatives.
<p><strong>Solution:</strong> From the graph of <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>), <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is continuous for all <span style="font-style: italic;">x</span> ∈ ℝ and <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is not differentiable at <span style="font-style: italic;">x</span> = +1, 0, 1, 3 (four points).</p><p>∴ Answers are (a) and (d).</p>
Correct Answer: A, D