I have a problem understanding second-order conditions at a critical point in finding critical points of two-variable functions.
Let's consider $f(x,y)$.
The first-order conditions are $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$. So the rate of change of $f$ in respect to both $x$ and $y$ is naught at a critical point.
The second-order conditions at a critical point that I have in my book are of the following form: A point (a,b) is a maximum if $f_{xx} f_{yy} - f_{xy}^2 > 0 $ and $f_{xx} < 0$. (For all my functions, $f_{xy}=f_{yx}$.)
What I don't understand is why the second-order conditions have to be so complicated. I think if $f_{xx} > 0$ and $f_{yy} > 0$ (both second-order derivatives are positive), we have a minimum. If $f_{xx}<0$ and $f_{yy}<0$, we have a maximum. And if the signs of two second-order derivatives at (a,b) are different it is a saddle point. What am I missing?