Answer
$(-\infty,0),(0,\infty)$
Work Step by Step
$r(t) =t^2 \textbf i + t^3 \textbf j \\
r'(t) = 2t\textbf i + 3t\textbf j\\
r'(t) = 0\textbf i + 0\textbf j \text{ for } t=0$
Therefore, the curve is smooth on the intervals $(-\infty,0),(0,\infty)$.
