The Derivative · Superchallenge problem

The function $f(x)=x^2|x|$ (that is, $f(x)=x^3$ for $x\ge0$ and $f(x)=-x^3$ for $x<0$) is differentiable some number of times at $x=0$ before the pattern breaks. Find the largest $n$ for which $f^{(n)}(0)$ exists.