Define the function .

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

Define the function

y=(|x|+2)^{3}

. (a) Rewrite

y

as a piecewise-defined function that does not involve the absolute value symbol. (b) For

x<0

and for

x>0

, compute

\dfrac{dy}{dx}

and present each result in simplest form. (c) Determine whether

\dfrac{dy}{dx}

exists at

x=0

. Provide a detailed justification based on one-sided limits of the derivative or an equivalent argument.