Max-min in Several Variables

One of the important uses of multivariable calculus is in finding optimal values of functions of several variables.  We begin with a study of  2-variable functions where we can see what is largest and smallest on an explicit graph.  Notice on the explicit graph at the left that the high and low points occur on the edge of the graph.  An important topic in the course, called "LaGrange multipliers" gives an analytical condition needed for a boundary optimum.  The "gradient vector" of a function [Graphics:../Images/index_gr_40.gif]  is computed by partial derivatives, [Graphics:../Images/index_gr_41.gif] and it tells us the direction of [Graphics:../Images/index_gr_42.gif]change to move in order to make [Graphics:../Images/index_gr_43.gif]  increase at the fastest rate.

Click on the picture below, drag to make it bigger, and see if you can see what property the gradient arrows shown on the right have to have at maxima and minima.  (Once you have a little practice with gradients this will be easy, but you might be able to tell now just by comparing the two figures.)

[Graphics:../Images/index_gr_44.gif]

Mathematica will help us see the theory so we can apply it analytically as in the next example.

Analytical Example: [Graphics:../Images/index_gr_45.gif]  on [Graphics:../Images/index_gr_46.gif]

[Graphics:../Images/index_gr_47.gif]  has no interior critical points, so the max and min, which exist by the Extreme Value Theorem, must lie on the boundary circle.  The only boundary critical solutions are [Graphics:../Images/index_gr_48.gif], where [Graphics:../Images/index_gr_49.gif], the max, and [Graphics:../Images/index_gr_50.gif], where [Graphics:../Images/index_gr_51.gif], the min.

[Graphics:../Images/index_gr_52.gif]

Animated Gradient Ascent:

Later in the course we study vector fields and the flows they produce.  The idea is to move in the direction of a given vector.  Since the gradient points in the direction of fastest increase in our function, if we move in the direction of the gradient, we should head toward a maximum.  (At least a "local" one.)  This is illustrated in the next figure and animated in the link below.

[Graphics:../Images/index_gr_53.gif]

Load animation 1 showing steepest ascent moving in the direction of the gradient.
Load animation 2 showing steepest descent in the direction opposite the gradient.

Converted by Mathematica      May 7, 2001