Maxima and Minima from Contour Maps
Local Maxima and Minima of F(x,y)The input values (a,b) where a function F(x,y) has a local minimum output value or a local maximum output value is called a critical point.
Critical points live in the domain of the function F(x,y). So do the contours of F(x,y). Therefore, it seems reasonable to look for maxima and minima in the contour diagrams.
The general process in finding local maxima and minima from contours is to graph the contour diagram of a function F(x,y), and then "look for the maxima and minima" by looking for concentric circular-type regions culminating in a point - a critical point. Here we see the comparison between the 2D contour graph and the 3D surface graph, and its maxima and minima for this region.
What is the 2D Gradient Doing at a Critical Point?I am not going to tell you quite yet. Let's let you figure this one out. Our goal will be to complete the following theorem statements:
#1. If (a,b) is a critical point for the function F(x,y), then the gradient of F evaluated at (a,b) is ...
#2. If (a,b) is a critical point for the function F(x,y), then the partial derivatives of F evaluated at (a,b) are ...
#3. If (a,b) is a critical point for the function F(x,y), then the tangent plane at (a,b) is ...