= Solution
Put $\xi=x$ and $\eta=y-1$. The <polynomial> becomes
$$
g=-\xi^3(1+\eta)^2(\xi+\eta)
=-\xi^4-\xi^3\eta+O\bigl((|\xi|+|\eta|)^5\bigr).
$$
There are no terms of degree one or two, so the <gradient> and the entire <Hessian matrix> vanish at $(0,1)$. It is a <critical point> with singular <Hessian matrix>. \b[The lowest nontrivial approximation is]
$$
\boxed{g(x,y)\sim-x^3\bigl(x+(y-1)\bigr),}
$$
understood as the fourth-degree leading term rather than a uniform ratio asymptotic along its zero directions.
On the path $\eta=0$ the exact value is $-\xi^4<0$ for $\xi\ne0$. On the path $\eta=-2\xi$ it is $\xi^4(1-2\xi)^2>0$ for sufficiently small nonzero $\xi$. Both paths approach the point, where $g=0$. \b[It is neither a <local maximum> nor a <local minimum>; it is a degenerate saddle point of a scalar function.] The <higher-order saddle test> supplies the conclusion the singular <Hessian matrix> cannot supply.
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