Question

If $f\left(x\right)=\{\begin{array}{c}3,x<0\\ 2x+1,x\ge 0\end{array}$, then

• A. both $f\left(x\right)$ and $f\left(\left|x\right|\right)$ are differentiable at $x=0$
• B. $f\left(x\right)$ is differentiable but $f\left(\left|x\right|\right)$ is not differentiable at $x=0$
• C. $f\left(x\right)$ is not differentiable at $x=0$
• D. both $f\left(x\right)$ and $f\left(\left|x\right|\right)$ are not differentiable at $x=0$

Answer 1

The correct option is C $f\left(x\right)$ is not differentiable at $x=0$

$\underset{x\to 0^{+}}{lim}f\left(x\right)$

$=\underset{x\to 0^{+}}{lim}2x+1$

$=1$

$\underset{x\to 0^{-}}{lim}f\left(x\right)$

$=\underset{x\to 0^{-}}{lim}3$

$=3$

Hence

$LHL\ne RHL$ at 0.

Hence the function is continuous at $x=0$.

Hence the function is not differentiable at $x=0$.