Question

Write the points where f (x) = |log_e x| is not differentiable.

Answer 1

Given:  

`f(x) |log_e x| = {(-log_e,x, ,0 <x<1,),(log_e,x, ,x≥1,):}`

Clearly 

\[f(x)\]  is differentiable for all 

\[x > 1\]  and for all 

\[x < 1\] . So, we have to check the differentiability at 

\[x = 1\]

We have,
(LHD at x = 1) 

\[\lim_{x \to 1^-} \frac{f(x) - f(1)}{x - 1} \]

\[= \lim_{x \to 1^-} \frac{- \log x - \log 1}{x - 1}\]
\[ = - \lim_{x \to 1^-} \frac{\log x}{x - 1}\]
\[ = - \lim_{h \to 0} \frac{\log (1 - h)}{1 - h - 1}\]
\[ = - \lim_{h \to 0} \frac{\log (1 - h)}{- h} \]
\[ = - 1\]

(RHD at x=1)

\[\lim_{x \to 1^+} \frac{f(x) - f(1)}{x - 1}\]

\[= \lim_{x \to 1^+} \frac{\log x - \log 1}{x - 1}\]
\[ = \lim_{x \to 1^+} \frac{\log x}{x - 1}\]
\[ {= \lim_{h \to 0}}_{} \frac{\log (1 + h)}{1 + h - 1}\]
\[ = \lim_{h \to 0} \frac{\log (1 + h)}{h}\]
\[ = 1\]

Thus, (LHD at x =1) ≠ (RHD at x =1)
So, 

\[f(x)\]  is not differentiable at 

\[x = 1 .\]