Question

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

• A. $f^{\prime }\left(a^{+}\right)=\varphi \left(a\right)$
• B. $f^{\prime }\left(a^{-}\right)=-\varphi \left(a\right)$
• C. $f^{\prime }\left(a^{+}\right)=f^{\prime }(a-)$
• D. None of these

Answer 1

Answer: (A), (B)

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

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

$f^{\prime }\left(a^{+}\right)=\varphi \left(a\right),f^{\prime }\left(a^{-}\right)=-\varphi \left(a\right)$