The two positive real numbers with the given sum whose product is a maximum can be found by using the AM-GM inequality. According to the AM-GM inequality, the arithmetic mean of two positive real numbers is always greater than or equal to their geometric mean. Equality occurs when the two numbers are equal.
Therefore, to maximize the product, we want the two numbers to be as close to each other as possible. So, we set the two numbers to be equal to x and x, where x is equal to the half of the sum, 30/2 = 15.
The two positive real numbers with the given sum whose product is a maximum are:
15, 15