Limits, Continuity & Differentiability
Non-differentiability of Greatest Integer Function
Grade 12
Question:
<p><strong>Ex. 12</strong> If $f(x) = [2 + 5|n|\sin x]$, where $n \in \mathbb{I}$ has exactly 9 points of non-derivability in $(0, \pi)$, then possible values of $n$ are (where $[x]$ denotes greatest integer function)</p>
<p>(a) $\pm 3$</p>
<p>(b) $\pm 2$</p>
<p>(c) $\pm 1$</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The greatest integer function has discontinuities (and hence non-differentiability) at integer values. The number of such points in the range of $5|n|\sin x$ over $(0,\pi)$ determines the total non-derivability points.
<p><strong>Solution:</strong> We have $[2 + 5|n|\sin x] = 2 + [5|n|\sin x]$.</p><p>Let $y = 5|n|\sin x$.</p><p>The number of points of non-derivability = $2(5|n| + 1) - 1 = 10|n| + 1$</p><p>According to the question, $10|n| + 1 = 9$</p><p>$10|n| = 8$</p><p>$|n| = 1$</p><p>$\therefore n = \pm 1$</p>
Correct Answer: C