If f(x)=|x-a| (x), where (x) is continuous function, then : - Vidyadip

MCQ

If $\text{f(x)}=|\text{x}-\text{a}|\ \phi\ (\text{x}),$ where $\phi(\text{x})$ is continuous function, then :

A
$\text{f}\ '(\text{a}^+)=\phi(\text{a})$
✓
$\text{f}\ '(\text{a}^-)=-\phi(\text{a})$
C
$\text{f}\ '(\text{a}^+)=\text{f}'(\text{a}^-)$
D
None of these

✓

Answer

Correct option: B.

$\text{f}\ '(\text{a}^-)=-\phi(\text{a})$

Given that $\text{f(x)}=|\text{x}-\text{a}|\ \phi\ (\text{x}),$ where $\phi(\text{x})$ continuous function.
$|\text{x}-\text{a}|$
$\Rightarrow\text{x}-\text{a}$ if $\text{x}-\text{a} > 0$
$|\text{x}-\text{a}|$
$\Rightarrow-(\text{x}-\text{a})$ if $\text{x}-\text{a} < 0$
By definition of continuity,
$\text{f}'(\text{a})= \lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{a}+\text{h})-\text{f(a)}}{\text{h}}$
Hence, $\text{f}(\text{a}^+)=\phi(\text{x})$ and $\text{f}'(\text{a}^-)=-\phi(\text{x})$

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