Browse Questions

# Find the set of all points where the function f(x) = 2x |x| is differentiable.

Toolbox:
• Write the function $2x | x |$ as $f(x)= \left\{ \begin{array}{1 1} 2x^2, & \quad \;x\geq0 \\ -2x^2, & \quad \;x<0 \end{array}. \right.$
• LHD = RHD = $\lim\limits_{x \to 0} \: \large\frac{f(x)-f(0)}{x-0}$
• Check only at breaking point
$f(x) = 2x^2\: \: if \: x \geq 0$
$\: \: \: \: \: \:\: \: \: = -2x^2\: \: if \: x < 0$

LHD = $\lim\limits_{x \to 0} \: \large\frac{2x^2-0}{x-0}=0$
RHD = $\lim\limits_{x \to 0} \: \large\frac{-2x^2-0}{x-0}=0$

Ans $\Rightarrow f$ is differentiable at every real number i.e., $(\infty, \infty)$

edited Mar 26, 2013