Question

The set of points where the function $f(x)=x|x|$ is differentiable is

• A. $\left(-\infty ,\infty \right)$
• B. $\left(-\infty ,0\right)\cup \left(0,\infty \right)$
• C. $\left(0,\infty \right)$
• D. $\left[0,\infty \right]$

Answer 1

Answer: (A)

We have, $f(x)=\{\begin{array}{cl}{x}^2,& x\ge 0\\ -x^2,& x<0\end{array}$
Clearly, $f(x)$ is differentiable for all $x>0$ and for all $x<0.$
So, we check the differentiability at $x=0$
Now, (RHD at $x=a$ )
$={\left(\frac{d}{dx}\left(x^2\right)\right)}_{x=0}=(2x{)}_{x=0}=0$
$\therefore (LHD\text{ at }x=0)={\left(\frac{d}{dx}\left(-x^2\right)\right)}_{x=0}=(-2x{)}_{x=0}=0$
$(LHD$ at $x=0)=(RHD$ at $x=0)$
So, $f(x)$ is differentiable for all $x$ ie, the set of all points where $f(x)$ is differentiable is $(-\infty ,\infty )ie,R$