Types of critical points Let us now consider that there are more than maximum and minimum. Therefore, remember that we considered the example of x2 + y2, which has a critical point but is obviously a minimum. Of course this could be a local minimum because it could be that in a more complicated function there is indeed a minimum here. But elsewhere, the function drops to a lower value. So we call that just a local minimum to say that it's a minimum if you stick to values that are close enough to that point. The function also has a local maximum, but it is easy to locate because it occurs where the curve becomes steeper. However, there is another example of a critical point: the saddle point. It is a phenomenon that cannot be observed in single variable calculus, but it does exist. Here, we see the critical point at the origin is a saddle point because depending on which direction we look from it will either appear to be a maximum or minimum. If you look at the tangent plane to this graph, you'll see that it is horizontal at the origin. You have a mountain pass, where the ground is horizontal. But depending on which direction you go, you go up or down. So we say that if a point is neither a minimum or maximum, it's a saddle point.