Question

Assertion :$f\left(x\right)=e^{-\left|x\right|}$ is differentiable for all x Reason: $f\left(x\right)=e^{-\left|x\right|}$ is continuous for all x

• 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 D Assertion is incorrect but Reason is correct

$f\left(x\right)=\{\begin{array}{cc}{e}^{-x}& \text{if}x\ge 0\\ e^x& \text{if}x<0\end{array}$
Thus $f^{\prime }\left(x\right)=\{\begin{array}{cc}-e^{-x}& \text{if}x>0\\ e^x& \text{if}x<0\end{array}$
$\underset{h\to 0+}{lim}\frac{f(0+h)-f\left(0\right)}{h}=(-1)\underset{h\to 0+}{lim}\frac{e^{-h}-1}{-h}=-1$ and
$\underset{h\to 0-}{lim}\frac{f(0+h)-f\left(0\right)}{h}=\underset{h\to 0-}{lim}\frac{e^h-1}{h}=1$.
So $f$ is differentiable on $R\sim \left\{0\right\}$.
Since $g\left(x\right)=|x|$ and $h\left(x\right)=e^{-x}$ are continuous functions so $hog\left(x\right)=e^{-\left|x\right|}=f\left(x\right)$ is a continuous function.