Limits, Continuity & Differentiability
Parametric Equations and Differentiability
Grade 12
Question:
<p>Let <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) = |<span style="font-style: italic;">x</span> + 1|([<span style="font-style: italic;">x</span>] + [−<span style="font-style: italic;">x</span>]), where [·] denotes the greatest integer function, then which of the following statement(s) is/are correct?</p>
<p>(a) <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is continuous at <span style="font-style: italic;">x</span> = 1</p>
<p>(b) <span style="font-style: italic;">f</span>(<span style="font-style: italic;">x</span>) is derivable at <span style="font-style: italic;">x</span> = 1</p>
<p>(c) <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>(d) <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 \ {2}</p>
Step-by-Step Solution
Key Concept: Analyze parametric equations by considering different ranges of the parameter. Check continuity and differentiability at critical points where the parameter transitions.
<p><strong>Solution:</strong> The function involves parametric equations <span style="font-style: italic;">x</span> = 2<span style="font-style: italic;">t</span> − |<span style="font-style: italic;">t</span> − 1| and <span style="font-style: italic;">y</span> = 2<span style="font-style: italic;">t</span> − t²/|<span style="font-style: italic;">t</span>|. Analyzing for different ranges of <span style="font-style: italic;">t</span>:</p><p>When <span style="font-style: italic;">t</span> ≥ 0: <span style="font-style: italic;">x</span> = 3<span style="font-style: italic;">t</span> − 1/2 and <span style="font-style: italic;">y</span> = 1/9(<span style="font-style: italic;">x</span> + 1)²</p><p>When 0 ≤ <span style="font-style: italic;">t</span> ≤ 1: <span style="font-style: italic;">y</span> = 1/3(<span style="font-style: italic;">x</span> + 1)²</p><p>When <span style="font-style: italic;">t</span> > 1: <span style="font-style: italic;">y</span> = 3(<span style="font-style: italic;">x</span> − 1)²</p><p>Checking continuity at <span style="font-style: italic;">x</span> = −1 and <span style="font-style: italic;">x</span> = 2 confirms continuity. Non-differentiability occurs at <span style="font-style: italic;">x</span> = 2.</p><p>∴ Answers are (a) and (d).</p>
Correct Answer: A, D