Question

Let $f\left(x\right)=x\left|x\right|$ The set of points where $f\left(x\right)$ is twice differentiable be:

• A. $(-\infty ,\infty )$
• B. $(1,\infty )$
• C. $Z–\left\{0\right\}$
• D. $R-\left\{0\right\}$

Answer 1

The correct option is D

$R-\left\{0\right\}$

Explanation for the correct option

Finding the points at which $f\left(x\right)$ is differentiable twice.
The given function is,

$f\left(x\right)=\left\{\begin{array}{l}{x}^2,x⩾0\\ -x^2,x<0\end{array}\right.$

Also, $f\left(0\right)=0$

$f\text{'}\left(x\right)=\left\{\begin{array}{l}2x,x\ge 0\\ -2x,x<0\end{array}\right.$

Now, double differentiating

$f"\left(x\right)=\left\{\begin{array}{l}2,x⩾0\\ -2,x<0\end{array}\right.$

$f\text{'}\text{'}\left(x\right)$ is not continuous at $0$. Hence, $f\left(x\right)$ is not differentiable at $0$.
Therefore $f\left(x\right)$is not twice differentiable at $0$.
If we take this point out of the domain, the function will be twice differentiable at all places. So it is twice differentiable in $R-\left\{0\right\}$.
Hence, the correct answer is Option (D).