find two positive integers such that the sum of the first number and 4 times the second number is 1000 and the product of the numbers is as large as possible

Answer 1

let the second number be x,
let the first number be y
so y + 4x = 1000, ---> y = 1000-x

Product = xy
= x(1000-4x)
= 1000x - 4x^2

find d(product)/dx
set it equal to zero and solve for x
comes out very nice.