But notice what happens at a maximum point, a place where the graph is at the largest value. We also call a point a critical point if one or more of the partial derivatives doesn't exist there.
Critical points are the places to look for a maximum or minimum of the function.
There is a new phenomenon for functions of more than one variable, though. All those things can happen for functions of several variables as well, but the point can also be a saddle point, which is not the same as a plateau. It's stable, not just a bare leveling off, but any nearby surface will also have such a saddle point. There is a test to see what kind of critical point it is, essentially the new version of the second derivative test. If the region, the domain of the function, has a boundary, then you have to look for critical points on the interior, and separately on the boundary.