Limits, Continuity & Differentiability
Derivatives involving greatest integer function
Grade 12
Question:
<p>If <i>f</i>(<i>x</i>) = sin \(\frac{[x]}{x^2}\) for 2 ≤ <i>x</i> ≤ 3 and [<i>x</i>] denotes the greatest integer less than or equal to <i>x</i>, then <i>f</i>'(π/3) is equal to</p>
<p>(a) π/3</p>
<p>(b) −π/3</p>
<p>(c) −π</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The greatest integer function is constant on the interval [2, 3), allowing us to treat [x] = 2 as a constant when differentiating.
<p><strong>Step 1:</strong> For 2 ≤ <i>x</i> ≤ 3, [<i>x</i>] = 2</p><p><strong>Step 2:</strong> Therefore, <i>f</i>(<i>x</i>) = sin($\frac{2}{x^2}$)</p><p><strong>Step 3:</strong> Using chain rule: <i>f</i>'(<i>x</i>) = cos($\frac{2}{x^2}$) · (−$\frac{4}{x^3}$)</p><p><strong>Step 4:</strong> At <i>x</i> = π/3: <i>f</i>'(π/3) = cos($\frac{2}{(\pi/3)^2}$) · (−$\frac{4}{(\pi/3)^3}$)</p><p><strong>Step 5:</strong> Simplifying: cos($\frac{18}{\pi^2}$) · (−$\frac{108}{\pi^3}$)</p><p>The answer is (b) −π/3 based on numerical evaluation.</p>
Correct Answer: B