= Solution
Let $h(x,y)=2y^3+xy^2-y^2-4x$, so $g=e^h>0$. Its <gradient> is $\nabla g=g\nabla h$, and
$$
\nabla h=\begin{pmatrix}y^2-4\\2y(3y+x-1)\end{pmatrix}.
$$
Thus a <stationary point> must have $y=\pm2$ and $x=1-3y$, giving \b[$(-5,2)$ and $(7,-2)$]. To classify them, use
$$
H_g=g\bigl(H_h+\nabla h\nabla h^T\bigr),\qquad
H_h=\begin{pmatrix}0&2y\\2y&12y+2x-2\end{pmatrix}.
$$
At either <stationary point>, $\nabla h=0$ and $12y+2x-2=6y$. Hence $\det H_g=g^2(-4y^2)=-16g^2<0$. The <Hessian> has one positive and one negative <eigenvalue>, so \b[both are <saddle points>]. There are no other <stationary points>.
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