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Question: Let f(x) = (x + \|x\|) \|x\|. Then for all x...

Let f(x) = (x + |x|) |x|. Then for all x

A

f is continuous

B

f is differentiable for some x

C

f ' is continuous

D

All the above

Answer

All the above

Explanation

Solution

f(x) = (x + |x|) |x|

= {2x2x>00x≤0 \left\{ \begin{matrix} 2x^{2} & x > 0 \\ 0 & x \leq 0 \end{matrix} \right.\

continuous

f'(0 + h) = 4x & f'(0 - h) = 0 \\ at\mspace{6mu} x = 0\mspace{6mu}\mspace{6mu} = 0 & \end{matrix}$$ f, differentiable f ′(x) = $\left\{ \begin{matrix} 4x & x > 0 \\ 0 & x \leq 0 \end{matrix} \right.\ $ continuous