A real estate office manages $50$ apartments in a downtown building. when the rent is $\$900 $ per month, all units are occupied.
for every $\$25 $ increase in rent, one unit becomes vacant. on average, all units require $\$75$ in maintenance and repairs each month. how much rent should the real estate office charge to maximize profit?
On
If the correct answer is $1100$ or $1125$, maybe the function that should be maximizing is $$ \max[(900+25x)(50-x)-75(50-x)].\tag1 $$ I guess since $x$ units are vacant so the vacant apartments do not need maintenance and repair each month, then there are only $(50-x)$ units that need maintenance and repair each month. Although the question said 'all units', maybe there is a 'mistake'. Assuming that the equation $(1)$ that should be maximizing. Hence $$ \begin{align} \frac{d}{dx}((900+25x)(50-x)-75(50-x))&=0\\ 25(50-x)-(900+25x)+75&=0\\ 50x&=25(50)-900+75\\ x&=\frac{25(50)-900+75}{50}\\ &=\frac{17}{2}\\ &=8.5. \end{align} $$
Thus, we have $8$ or $9$ units are vacant and the rent should be charged by the real estate office to maximize profit is $$ 900+25(8)=\boxed{\color{blue}{1100}} $$ or $$ 900+25(9)=\boxed{\color{blue}{1125}} $$
$$\large\color{blue}{\text{# }\mathbb{Q.E.D.}\text{ #}}$$
We have the problem $max$ $(900 + 25x)(50 - x) - 3750$. So $41250 + 350x - 25x^{2} = 0$. Now simply apply the first derivative test: $350 - 50x = 0$, so $x = 7$. And so $900 + 25 * 7 = 1075$ is your answer.
As the coefficient of $x^{2}$ is negative, the function is maximized at $x = 7$. You can alternatively check the change of derivative around $x = 7$ to confirm.