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

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  • 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)$

 

answered Mar 12, 2013 by thanvigandhi_1
edited Mar 26, 2013 by thanvigandhi_1
 

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