Showing a given parameterized contour is smooth

Question

Show that the range of the function $z(t) = t^3 +it^6$, $-1 \leq t \leq 1$ is a smooth curve, even though the given parameterization is not admissable.

I'm not entirely sure how to solve this problem, does any one have any hints?

Answer 1

Note that the curve is the same as $z(u)=u+iu^2$ for $-1\le u\le1$.

Next notice that $z'(u)=1+2iu$ does not vanish for any $u\in[-1,1]$.