(820,#41) Find three positive numbers whose sum is 100

(820,#41) Find three positive numbers whose sum is 100 and whose product is a maximum.
Solution:

If the product is a maximum, then :

Substituting into B),

Thus, we have the critical points (0,0), (100,0), (0,100), (100/3,100/3)
Using the Second Derivative Test,

Thus, (0,0), (100,0), (0,100) are saddle points and is a maximum.
Therefore, the product is a maximum when .

by:sh