Thread: optimization w/ implicit differentiation

I need help in the setup of this word problem: Find two positive integers whose sum is 50 and whose product is as large as possible.
I set it up as $\displaystyle x+y=50$ and $\displaystyle xy=$as large as possible.
I know that these are usually done by solving for one of the variables, substituting it into the other problem, taking the derivative, finding the absolute max, blah blah, but I'm confused because the second equation has an indeterminate answer. Any help on how to set this up would be awesome!

To wrap this up: You maximize $\displaystyle xy$ s.t. $\displaystyle x+y = 50$.

Thus $\displaystyle x = 50 - y$ and you can maximize $\displaystyle y(50-y) = 50y - y^2$ and the solution to this is determinate since it is implicitly constrained by your first equation.

Ah, so i get 50x-x^2 f'(x)=50-2x x=25 which makes y=25 as well which is correct! Thanks a ton!