What are the two positive integers whose sum is 200 and whose product is a maximum?
To find the two positive integers whose sum is 200 and whose product is a maximum, we can use a mathematical approach called optimization.
Let's assume the two positive integers are x and y, where x + y = 200. We need to find the values of x and y that maximize the product xy.
To solve this problem, we can use the concept of AM-GM inequality. According to this inequality, the arithmetic mean (AM) of any set of positive numbers is always greater than or equal to their geometric mean (GM). In other words:
(x + y)/2 ≥ √(xy)
Now, substituting x + y = 200 into the inequality, we have:
200/2 ≥ √(xy)
Simplifying, we get:
100 ≥ √(xy)
Now, let's square both sides of the inequality:
(100)² ≥ (√(xy))²
10,000 ≥ xy
This means that the maximum value of xy is 10,000, and it occurs when x = y = 100.
Therefore, the two positive integers whose sum is 200 and whose product is a maximum are 100 and 100.