= Solution
Differentiating the projected parameterization gives
$$
\boxed{J=\frac{\partial(x,y)}{\partial(s,t)}=\det\begin{pmatrix}1&1\\1+2s&-1\end{pmatrix}=-2(s+1).}
$$
It vanishes exactly when $s=-1$. The image is the tangency half-line from part (iii), $x=t-1$, $y=-t$, $t\geq0$. This is a <fold caustic>: the parameterized surface itself remains regular, but its projection ceases to be locally invertible, so it cannot there be expressed as a differentiable single-valued graph over the plane.
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