Find two numbers whose sum is 15 and whose product is as large as possible.

If one number is x the other is 15-x so their product is 15x-x².
This can be written -(x²-15x+225/4)+225/4=(15/2)²-(x-15/2)².
The largest value of this expression is when x=15/2=7.5 so the other number is also 7.5.
Let’s pick two other numbers and check their product: 7 and 8, product=56 but the product of 7.5 and 7.5=56.25 which is larger, therefore the two numbers are 7.5 and 7.5.