Limits Continuity And Differentiability Question 33
Question: Let $ f:R\to R $ be defined as $ f(x)=sin(| x |) $ Which one of the following is correct?
Options:
A) f is not differentiable only at 0
B) f is differentiable at 9 only
C) f is differentiable everywhere
D) f is non-differentiable at many points
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] Given function is : $ f(x)=\sin | x | $
$ = \begin{matrix} \sin (x), & x\ge 0 \\ \sin (-x), & x<0 \\ \end{matrix} . $
$ = \begin{matrix} \sin ,x, & x\ge 0 \\ -\sin ,x, & x<0 \\ \end{matrix} . $ LHD at $ x=0=\underset{h\to 0}{\mathop{\lim }},\frac{f(0-h)-f(0)}{0-h-0} $
$ =\underset{h\to 0}{\mathop{\lim }},\frac{f(0-h)-f(0)}{-h}=\underset{h\to 0}{\mathop{\lim }},\frac{-\sin (-h)-0}{-h}=-1 $
RHD at $ x=0=\underset{h\to 0}{\mathop{\lim }},\frac{f(0+h)-f(x)}{0+h-0} $
$ =\underset{h\to 0}{\mathop{\lim }},\frac{f(0+h)-f(0)}{h}=\underset{h\to 0}{\mathop{\lim }},\frac{\sin (h-0)}{h}=1 $
$ LHD\ne RHD $
$ f(x) $ is not differentiable at $ x=0 $ .
BETA
