Begin2.DVI

The problem of determining the relative maximum and minimum values of a

function of two variables is now considered. In the discussions that follow, note

that the problem of determining the maximum and minimum for a function of two

variables is reduced to the simpler problem of finding the maximum and minimum

of a function of a single variable.

If (x 0 , y 0 )is a critical point associated with the surface z=z(x, y ),then one can

slide the free vector given by eˆα= cos αˆe 1 + sin αˆe 2 to the critical point and construct

a plane normal to the plane z= 0,such that this plane contains the vector ˆeα.This

plane intersects the surface in a curve. The situation is depicted graphically in the

figure 7-14

At a critical point where ∂x∂z = 0 and ∂z∂y = 0,the directional derivative satisfies

dz

ds = grad z·eˆα=

∂z

∂x cos α+

∂z

∂y sin α= 0

for all directions α.

Figure 7-14. Curve of intersection with plane containing ˆeα.

Here the directional derivative represents the variation of the surface height z

with respect to a distance sin the ˆeα direction. (i.e., measure the rate of change