Profit maximization problem using optimization An apartment complex on Ferenginar with $250$ units currently has $193$ occupants. The current rent for a unit is $965$ slips of Gold-Pressed Latinum. The owner of the complex knows from experience that he loses one occupant every time he raises the rent by $2.5$ slips of Latinum. What should be our recommendation for the optimal rent? The equation I got was $(193-x)\cdot(965+2.5x)$. I tried to maximize it and got $x=-96.5$. The optimal rent is $965+2.5(-96.5)=723.75$. This answer is not correct? Where did I make a mistake Thanks

You did nothing wrong!

Notice that the equation of the total rent with respect to $x$ is a parabola. That means as $x$ goes from $0$ to $-96.5$, the total rent is increasing. However, our maximum occupancy is $250$ residents, so the rent is maximized at $x=-57$.