Differentiation Question 275
Question: If $ f(x)=|x|, $ then $ f’(0)= $
[MNR 1982]
Options:
A) 0
B) 1
C) x
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
$ f(x)=|x|, $ we have $ f(0)=|0|=0 $
$ f(0+0)=\underset{h\to 0}{\mathop{\lim }}|0+h|=0 $ and $ f(0-0)=\underset{h\to 0}{\mathop{\lim }}|0-h|=0 $
$ Rf’(0)=\underset{h\to 0}{\mathop{\lim }}\frac{f(0+h)-f(0)}{h}=\underset{h\to 0}{\mathop{\lim }}\frac{|h|-0}{h} $
$ =\underset{h\to 0}{\mathop{\lim }}\frac{h}{h} $ (h being positive)=1 $ Lf’(0)=\underset{h\to 0}{\mathop{\lim }}\frac{f(0-h)-f(0)}{-h}=\underset{h\to 0}{\mathop{\lim }}\frac{|h|-0}{-h} $
$ =\underset{h\to 0}{\mathop{\lim }}\frac{h}{-h}(h $ being positive) = -1.
$ \therefore Rf’(0)\ne Lf’(0) $ . The function f is not differentiable.
BETA
