Find 2 positive numbers that satisfy the requirements that the sum of the first and twice the second is 100 and the product is a maximum.

50 and 25...

Suppose the numbers to be x and y.

x+2y=100
then, y=1/2(100-x)

Let xy=z
Convert the left equivalent into x-terms and we get,
x/2(100-x) = z
To get the value of x when z is maximum, you need to differentiate the above equation and equal it to zero. That is,

1/2(100-2x) = 0 and we get x=50 and so, y=25