The sum of two positive numbers is 10. Find the numbers if their product is to be a maximum

x+y = 10

p = xy = x(10-x) = 10x-x^2
dp/dx = 10-2x
max p at x=5

what a surprise...

To find the numbers, let's suppose the two positive numbers are 'x' and 'y'.

Given that the sum of the two numbers is 10, we can write the equation:
x + y = 10

To find the numbers when their product is a maximum, we need to apply the concept of the maximum product property. According to this property, to obtain the maximum product between two numbers given their sum, they should be as close to each other as possible.

In this case, since the two numbers are positive and their sum is 10, the numbers should be equal or close to equal for their product to be maximized.

So, let's assume x = y (both numbers are equal) and substitute it in the equation:
x + x = 10
2x = 10
x = 10/2
x = 5

Hence, the two positive numbers are both 5 when their product is maximized.