Question

Assertion :$f\left(x\right)=e^{\left|x\right|}$ is non differentiable at $x=0$. Reason: L.H.D. of $f\left(x\right)$ at $x=0\ne$R. H. D.of $f\left(x\right)$ at $x=0$.

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right)=e^{\left|x\right|}=\{\begin{array}{cc}{e}^x& \text{if}x\ge 0\\ e^{-x}& \text{if}x<0\end{array}$
$\Rightarrow$ $f^{\prime }\left(x\right)=\{\begin{array}{cc}{e}^x& \text{if}x\ge 0\\ -e^{-x}& \text{if}x<0\end{array}$
$\Rightarrow$ $f^{\prime }\left(0^{+}\right)=e^0=1=$R.H.D.
and $f^{\prime }\left(0^{-}\right)=-e^{-0}=-1=$L.H.D.
Hence, option 'A' is correct.