Find the maximum and minimum product of 2 numbers who sum is 5

max product: (5/2)^2 = 25/4

min product: 0*5 = 0

assuming x >= 0

allowing negative values means there is no minimum product. For example, -100 * 105 = -10500.

y = x(5-x) = 5x - x^2

y' = 5-2x
y'=0 when x = 5/2

To find the maximum and minimum product of two numbers whose sum is 5, we need to determine the two numbers first.

Let's assume the two numbers are x and y such that x + y = 5.

To find the maximum product, we want to maximize the difference between the two numbers. This means we should choose x as close to 5 as possible and y as close to 0 as possible.

So, let x = 4 and y = 1. In this case, the maximum product will be attained with x and y.

Maximum product: x * y = 4 * 1 = 4.

To find the minimum product, we want to minimize the difference between the two numbers. This means we should choose x and y as close to each other as possible.

So, let x = 2.5 and y = 2.5. In this case, the minimum product will be attained with x and y.

Minimum product: x * y = 2.5 * 2.5 = 6.25.

Therefore, the maximum product of two numbers whose sum is 5 is 4, and the minimum product is 6.25.

To find the maximum and minimum product of two numbers whose sum is 5, let's consider the possible scenarios:

1. When both numbers are positive:
In this case, since both numbers are positive and their sum is 5, one number must be greater than 2.5 (half of 5) and the other number must be smaller than 2.5. Therefore, the maximum product occurs when both numbers are equal to 2.5, resulting in a maximum product of (2.5) x (2.5) = 6.25. The minimum product occurs when one number is close to zero and the other is close to 5, resulting in a minimum product very close to zero.

2. When one number is positive and the other is negative:
Here, we have two cases to consider: when the positive number is greater than 2.5 and when the negative number is greater than -2.5.

- When the positive number is greater than 2.5:
Since the positive number must be greater than 2.5 and the negative number must be smaller than -2.5, the maximum product occurs when both numbers are equal to 2.5, resulting in a maximum product of (2.5) x (2.5) = 6.25. The minimum product occurs when the positive number is close to 5 and the negative number is close to -5, resulting in a minimum product of (5) x (-5) = -25.

- When the negative number is greater than -2.5:
In this case, the maximum product occurs when one number is close to -2.5 and the other number is close to 5, resulting in a maximum product of (-2.5) x (5) = -12.5. The minimum product occurs when both numbers are close to zero, resulting in a minimum product very close to zero.

Based on these scenarios, the maximum product is 6.25 and the minimum product is -25.