Functions Question 222
Question: The left-hand derivative of $ f(x)=
[x]\sin (\pi x) $ at $ x=k,k $ is an integer and $ [x] $ = greatest integer $ \le x,, $ is [IIT Screening 2001]
Options:
A) $ {{(-1)}^{k}}(k-1),\pi $
B) $ {{(-1)}^{k-1}}(k-1),\pi $
C) $ {{(-1)}^{k}}k\pi $
D) $ {{(-1)}^{k-1}}k,\pi $
Show Answer
Answer:
Correct Answer: A
Solution:
$ {f}’(k-0)=\underset{h\to 0}{\mathop{lim}},\frac{[k-h]\sin \pi (k-h)-[k]\sin \pi k}{-h} $ $ =\underset{h\to 0}{\mathop{lim}},\frac{{{(-1)}^{k-1}}(k-1)\sin \pi h-k\times 0}{-h} $ $ =\underset{h\to 0}{\mathop{lim}},\frac{{{(-1)}^{k-1}}(k-1)\sin \pi h}{-h} $ $ ={{(-1)}^{k}}.(k-1)\pi $ .
BETA
