Solve this

Question:

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

(a) $f\left(a^{+}\right)=\phi(a)$

(b) $f\left(a^{-}\right)=-\phi(a)$

(c) $f\left(a^{+}\right)=f\left(a^{-}\right)$

(d) none of these

Solution:

(a) $f^{\prime}\left(a^{+}\right)=\phi(a)$

(b) $f^{\prime}\left(a^{-}\right)=-\phi(a)$

Here, $f(x)=|x-a| \phi(x)$

$f^{\prime}\left(a^{+}\right)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}=\lim _{h \rightarrow 0} \frac{|h+a-a| \phi(a+h)-|a-a| \phi(a)}{h}=\lim _{h \rightarrow 0} \frac{h \phi(a+h)}{h}=\lim _{h \rightarrow 0} \phi(a+h)=\phi(a)$

Also,

$f^{\prime}\left(a^{-}\right)=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{h}=\lim _{h \rightarrow 0} \frac{|a-h-a| \phi(a-h)-|a-a| \phi(a)}{h}=\lim _{h \rightarrow 0} \frac{|-h| \phi(a-h)}{h}=\lim _{h \rightarrow 0}-\phi(a-h)=-\phi(a)$

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