Lets arrange pairs of numbers which add to
We notice that as the numbers get closer, the difference between them gets less, but their product increases.
The smallest difference will be when the two numbers are equal and their product will therefore be the biggest.
The numbers are
Let one number be
Then, the other number is
Let the product of the numbers be
Differentiating with respect to
The critical value is when
Let's make a variation chart
There is a maximum at
We confirm this by calculating the second derivative