Final answer:
To find three positive numbers with a maximal product given their sum, set all three numbers equal to one third of the sum, applying the geometric mean inequality.
Step-by-step explanation:
Finding Three Positive Numbers with Maximal Product
To find three positive numbers whose sum is a given value and whose product is a maximum, we can use the concept of the geometric mean. The geometric mean for any set of positive numbers is greater than or equal to the arithmetic mean, with equality holding only when all numbers are the same. Therefore, to maximize the product of three numbers with a fixed sum, those numbers must be equal. Let's call these three numbers X, Y, and Z. If the sum of these numbers is S, then we have X + Y + Z = S and the numbers we are looking for will all be S/3.
The product P of three equal numbers that sum to S will thus be (S/3) × (S/3) × (S/3), which is essentially (S/3)^3. This is the maximal product that can be achieved under these conditions.