Solution
= Solution
On $y=0$, the restricted function is $g(x)=x^4$, so $g'(x)=4x^3$ changes from negative to positive at zero. Hence
$$
\boxed{x=0,\quad (x,y)=(0,0),\quad g=0:\text{strict minimum along }y=0.}
$$
The restricted second derivative vanishes there, but the quartic term determines the minimum. This does not contradict the two-dimensional saddle: values decrease along other directions, including the diagonal near the origin.