From the following relation:
How can we conclude the following rules:
(i) Minima if both $f_{xx}$ and $f_{yy}$ are positive and $(f_{xy})^2 < f_{xx} f_{yy}$,
(ii) Maxima if both $f_{xx}$ and $f_{yy}$ are negative and $(f_{xy})^2 < f_{xx}f_{yy}$,
(iii) Saddle points if $f_{xx}$ and $f_{yy}$ have opposite signs or $(f_{xy})^2 > f_{xx}f_{yy}$.
Hints: (i) In this case the terms on the right are all positive. (ii) Similar to (i) but with different sign. (iii) If $f_{xx}$ is positive then if change in $y$ is $0$ then the difference on the left is positive. But if change in $x$ is $0$ then the difference on the left is negative. If $f_{xx}$ is negative apply a similar analysis of the problem to conclude there is a saddle. If $f_{xy}^{2}>f_{xx}f_{yy}$ but $f_{xx}$ and $f_{yy}$ have the same sign then a similar analysis on changes in $x$ and $y$ reveals a saddle. The cases may coindcide so paying close attention to sign and make one of the differences $\Delta x$ and $\Delta y$ equal to $0$ reveals a saddle again.