2nd derivative test (problem 2 on page 213): Assume $f_x(a,b)=f_y(a,b)=0,$
- If $f_{xx}>0, f_{yy}>0,$ and $f_{xx}f_{yy}> f_{xy}^2$, then $z =f(x,y)$ attains a local minimum at $(x,y)=(a,b)$.
- If $f_{xx}<0, f_{yy}<0,$ and $f_{xx}f_{yy}> f_{xy}^2$, then $z =f(x,y)$ attains a local maximum at $(x,y)=(a,b)$.
- If $f_{xx}f_{yy}< f_{xy}^2$, then $z =f(x,y)$ attains neither a local maximum nor a local maximum at $(x,y)=(a,b)$.