Answer 1: The Insurance Salesman

The children's ages are 2, 2, and 9.

Reasoning

There are only eight combinations of ages that give a product of 36 namely; {1,1,36}, {1,2,18}, {1,3,12}, {1,4,9}, {1,6,6}, {2,2,9}, {2,3,6}, and {3,3,4}. Now, the sum of the ages in each combination is unique except in the case of {1,6,6} and {2,2,9}. Hence, only under these circumstances would the agent require further information. Knowing that there is an eldest child confirms the ages as 2, 2, and 9.