Problem : Find 2 positive numbers whose product is 25 and whose sum is a minimum.Objective: S = x + y . The goal is to minimize S . Constraint: xy = 25 . Substitute constraint into objective:
=+ x; domain = (0,∞)
S'(x) = 0 when x = 5
Use second derivative to classify:
S''(5) > 0 , so S has a local min at x = 5 . However, notice thatS''(x) is always positive on the interval (0,∞) , so S is always concave up on that interval, which means that the local min is also the absolute min. Therefore, 5 and 5 are the positive numbers with the smallest sum whose product is 25.