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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
If $\text{f(x)}=|\log_\text{e}\text{x}|,$ then :
$(a)\ \text{f}\ '(1^+)=1$
$(b)\ \text{f}\ '(1^-)=-1$
$(c)\ \text{f}\ '(1)=1$
$(d)\text{f}\ '(1)=-1$
  • A
    $a$ and $b$
  • ✓
    $a$ and $d$
  • C
    $c$ and $b$
  • D
    None of these
✓

Answer

Correct option: B.
$a$ and $d$
$\text{f(x)}=|\log_\text{e}\text{x}|,=\begin{cases}-\log_\text{e}\text{x}, & \text{for}0 < \text{x} < 1\\\log_\text{e}\text{x}, & \text{for x}\geq1\end{cases}$
Differentiability at $x = 1,$
We have,
$\text{(LHL}$ at $x = 1)$
$=\lim\limits_{\text{x}\rightarrow1^{-}}\frac{\text{f(c)}-\text{f}(1)}{\text{x}-1}$
$=\lim\limits_{\text{x}\rightarrow1^{-}}\frac{-\log\text{x}-\log12}{\text{x}-1}$
$=\lim\limits_{\text{x}\rightarrow1^{-}}\frac{\log\text{x}}{\text{x}-1}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\log(1-\text{h})}{1-\text{h}-1}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\log(1-\text{h})}{-\text{h}}$
$=-1$
$\text{(RHL}$ at $x = 1)$
$=\lim\limits_{\text{x}\rightarrow1^{+}}\frac{\text{f(c)}-\text{f}(1)}{\text{x}-1}$
$=\lim\limits_{\text{x}\rightarrow1^{+}}\frac{\log\text{x}-\log1}{\text{x}-1}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\log(1+\text{h})}{\text{x}-1}$
$=\lim\limits_{\text{h}\rightarrow0}\frac{\log(1+\text{h})}{\text{h}}$
$=1$

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