Question

Assertion :$f\left(x\right)=\left|x\right|.\sin x$ is differentiable at $x=0$. Reason: If $f\left(x\right)$ is not differentiable and $g\left(x\right)$ is differentiable at $x=a$, then $f\left(x\right).g\left(x\right)$ will be differentiable at $x=a$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Assertion

$f\left(x\right)=\left|x\right|\sin x$

$f{}^{\prime }\left(x\right)=\frac{\left|x\right|}{x}\sin x+\left|x\right|\cos x$

$\left|x\right|$ and $\sin x$ are continuous, $f\left(x\right)$ is also continuous

$f{}^{\prime }\left(0^{-}\right)=0$

$f{}^{\prime }\left(0^{+}\right)=0$

$f{}^{\prime }\left(0^{-}\right)=f{}^{\prime }\left(0^{+}\right)=0$

$\therefore$ $f\left(x\right)$ is differentiable at $x=0$

Reason

if $f\left(x\right)$ is not differentiable and $g\left(x\right)$ is differentiable at

$x=a$ Then $f\left(x\right).g\left(x\right)$ is may (or) may not be differentiable at $x=a$

Reason is not correct.