This means that we are constantly walking downhill as we approach the point `(0,2)`, making `(0,2)` a local minimum point.
Note that the gradient vectors all point outward near the point `(0,2)`, regardless of approach.
This means that we are constantly walking uphill as we approach the point `(-2,0)`, making `(-2,0)` a local maximum point.
Note that the gradient vectors all point inward near the point `(-2,0)`, regardless of approach.
That's because your are going uphill as you approach the saddle point, but fall away downhill in another direction. The gradient vectors "flow" toward the saddle point in one direction, only to "flow" away in another direction.
Note the action of the gradient field near the suspected saddle points at `(0,0)` and `(-2,2)`.
Recall that the gradient vectors point in the direction of greatest increase of the function i.e., they point "uphill.".
Wonderful! Here are some important observations regarding Figure 9. Similar comments are in order for the local maximum that is marked in Figure 1. That is why this point is a local minimum, and not an absolute minimum. However, in its immediate locale or neighborhood, it is a lowest point. In the case of the local minimum, note that it is not the absolute lowest point on the surface, because there are other points on the surface that are lower still. In Figure 1, we've marked a local minimum and a local maximum on the surface. A local maximum is a point on the surface that is the highet point in its immediate neighborhood. We will investigate three types of extrema:Ī local minimum is a point on a surface that is the lowest point in its immediate neighborhood. In this activity, we will apply those visualizations to help determine extrema of multivariable functions of the form `f:R^2\to R`. In the activities Contour Maps in Matlab and The Gradient in Matlab, we developed visualizations of level curves and the gradient field.