Challenge Level

Plot the graph of the function $y=f(x)$ where $f(x) = x(x+|x|)$. Find the first and second derivatives of the function. Show that the first derivative exists at $x=0$ but that the second derivative does not exist at $x = 0$.

NOTES AND BACKGROUND

This is a very simple question but it requires an understanding of how to handle functions that are defined differently on different parts of their domains.