Methods from Local extrema

For a function $f(x,y)$:

1. calculate $f_x$ and $f_y$;
2. solve $$f_x(x,y)=0, \qquad f_y(x,y)=0;$$
3. retain the simultaneous solutions in the domain;
4. classify each candidate using local geometry or the Hessian test.

For each stationary point:

1. calculate the Hessian entries at the point;
2. compute $D=f_{xx}f_{yy}-f_{xy}^2$;
3. use the sign of $D$ and then the sign of $f_{xx}$ when $D>0$;
4. if $D=0$, inspect higher-order terms or compare the function along carefully chosen curves.