pg. 826, #9

Using Lagrange multipliers, find the minimum and maximum of subject to the constraint .

Solution:

The domain of f is the surface pictured below.

To continue, set . Then, for ,

Three cases arise:

Case 1: All three of x, y and z are not zero. Then


Consequently .

Case 2 : One of x, y , and z is zero and the other two are not. Say z = 0. Then
Consequently

Case 3:Two of x, y, and z are 0, one is not. Say y = z = 0. Then

Thus the function has a global maximum of at the eight points and a global minimum of 1 at the six points .