Applications of Derivatives
Differentiability of Piecewise Functions
Grade 12
Question:
<p><strong>Ex. 17:</strong> <strong>Statement I</strong> Let \(f(x) = x[x]\) and \([\cdot]\) denotes greatest integer function, when \(x\) is not an integral, then rule for \(f'(x)\) is given by \([x]\).</p><p><strong>Statement II</strong> \(f'(x)\) does not exist for any \(x \in \mathbb{Z}\).</p>
Step-by-Step Solution
Key Concept: Analyze the function $f(x) = x[x]$ piecewise on each interval between consecutive integers to find the derivative and check differentiability at integer points.
<p><strong>Solution:</strong> For $f(x) = x[x]$:</p><p>For $-1 \leq x < 0$: $f(x) = -x$, so $f'(x) = -1$ and $[x] = -1$</p><p>For $0 \leq x < 1$: $f(x) = 0$, so $f'(x) = 0$ and $[x] = 0$</p><p>For $1 \leq x < 2$: $f(x) = x$, so $f'(x) = 1$ and $[x] = 1$</p><p>For $2 \leq x < 3$: $f(x) = 2x$, so $f'(x) = 2$ and $[x] = 2$</p><p>Therefore $f'(x) = [x]$ for $x \notin \mathbb{Z}$.</p><p>At integer points, the left and right derivatives differ, so $f'(x)$ does not exist for $x \in \mathbb{Z}$.</p><p>Hence, both statements are true and Statement II is the correct explanation of Statement I.</p>
Correct Answer: A