Find three positive numbers whose sum is 100 and whoseproduct is a maximum

To solve this problem, we can use the concept of geometric mean. The geometric mean of a set of positive numbers is always less than or equal to the arithmetic mean. In other words, for any set of positive numbers, the product of the numbers is maximized when the numbers are equal.

Step 1: Let's denote the three numbers as a, b, and c. According to the problem, we have two equations:

a + b + c = 100 (since the sum of the numbers is 100)

We want to maximize a * b * c.

Step 2: To maximize the product a * b * c under the constraint a + b + c = 100, the numbers a, b, and c should be equal (or as close as possible to being equal).

Step 3: So, we divide 100 by 3 to get approximately 33.33.

Step 4: However, since we need three positive integers, we can't have decimal numbers. So, we take two numbers as 33 and one number as 34.

Step 5: Check: 33 + 33 + 34 = 100

So, the three numbers are 33, 33, and 34.