Answer
The two numbers are $125$ and $500$
Work Step by Step
Let $x$ and $y$ be the two numbers.
$x+4y = 1000$
$x = 1000-4y$
We can write an expression for the product:
$P = xy = (1000-4y)(y) = 1000y - 4y^2$
We can find the point where $P'(y) = 0$:
$P'(y) = 1000-8y = 0$
$8y = 1000$
$y = 125$
Note that $P''(y) = -8 \lt 0$
Since this graph is concave downward, $y=125$ must be the point where the product is a maximum.
We can find $x$:
$x = 1000-4(125) = 500$
The two numbers are $125$ and $500$
