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

Answer 1

The correct option is D $f^{\prime }\left(a^{-}\right)=-\varphi \left(a\right)$

$f\left(x\right)=\{\begin{array}{c}(a-x)\varphi \left(x\right)ifx<a\\ (x-a)\varphi \left(x\right)ifx>a\end{array}$
$f^{\prime }\left(x\right)=\{\begin{array}{c}(a-x){\varphi }^{\prime }\left(x\right)-\varphi \left(x\right)ifx<a\\ (x-a){\varphi }^{\prime }\left(x\right)+\varphi \left(x\right)ifx>a\end{array}$
Hence $f^{\prime }(a-)=-\varphi \left(a\right),f^{\prime }(a+)=\varphi \left(a\right)$