Question

Let $f\left(x\right)=|x-a|\varphi \left(x\right)$, where $\varphi$ is a continuous function and $\varphi \left(a\right)\ne 0$. Then

• A. $f^{\prime }(a+)=\varphi \left(a\right)$
• B. $f$ is differentiable at $x=a$
• C. $f^{\prime }(a+)=-\varphi \left(a\right)$
• D. $f^{\prime }(a-)=-\varphi \left(a\right)$

Answer 1

Answer: (A), (D)

The correct options are
A $f^{\prime }(a+)=\varphi \left(a\right)$
D $f^{\prime }(a-)=-\varphi \left(a\right)$

given $f\left(x\right)=|x-a|\varphi \left(x\right),\varphi$ is continuos function $\varphi \left(a\right)\ne 0$

differentiate both sides wrt $x$

$\frac{df\left(x\right)}{dx}=f^{\prime }\left(x\right)=\frac{|x-a|}{(x-a)}\varphi \left(x\right)+|x-a|{\varphi }^{\prime }\left(x\right)$

if $x=a+,x-a>0\therefore f^{\prime }(a+)=\varphi \left(a\right)$

if $x=a-,x-a<0\therefore f^{\prime }(a-)=-\varphi \left(a\right)$

$\therefore f$ is not differentiable but continuos.