Maximum Word Problem Two positive numbers who have a sum of 18, and the product of the first number and the square of the other is a maximum.

I got one number is 6, and the other is 12. Not sure if correct.

Any insight?
$x+y=18$. Let $P=xy^2$. You have to find $x$ and $y$ which make $P$ a maximum. $P=xy^2=(18-y)y^2=18y^2-y^3$. When $P$ is maximum $\frac{dP}{dy}=0$.

$\frac{dP}{dy}=36y-3y^2=y(36-3y)$. So, $\frac{dP}{dy}=0$ when $y=0$ or $y=12$. But when $y=0, P=0$ and when $y=12, P&gt;0$. Hence $P$ takes its maximum when $y=12$. so $x=6$
$x + y = 18$ , so $y = 18 - x$
Then you can rewrite $xy^{2}$ ... and you just have to find the maximum of a fonction only depending of $x$, with $0&lt;x&lt;18$ an integer.

By the way you can see that your answer is wrong by calculating $17^{2}$ !