# A market survey suggests that, on the average, one additional unit will remain vacant for each 3 dollar increase in rent. Similarly, one additional unit will be occupied for each 3 dollar decrease in rent. What rent should the manager charge to maximize?

Jan 24, 2016

There is not enough information provided to give a dollar value answer.

#### Explanation:

In order to give a numerical answer, we need to know what base we are increasing/decreasing rent from and what number of units are rented at that base rent.

Let $B$ = the base rent and
$N$ = the number of units occupied at rent $B$.

To maximize Revenue (which I assume is what we want to maximize), apply $\frac{3 N - B}{6}$ increments of $3 to the base rent. (If this is a positive number, increase the rent, if negative decrease it.) Let $k$be the number of $3 increments from the base rent, $B$.

The number of units occupied at rent $B + 3 k$ is $N - k$. (It is $N$ reduced by 1 per $k$.)

The Revenue will be:

$R \left(k\right) = \left(B + 3 k\right) \left(N - k\right)$

$= B N - B k + 3 N k - 3 {k}^{2}$

Maximize as usual. (Find and test the critical numbers -- or use your knowledge of quadratic functions)

$R ' \left(k\right) = - B + 3 N - 6 k$

$R '$ is never undefined and is $0$ at $k = \frac{3 N - B}{6}$

The second derivative test tells us that $R \left(\frac{3 N - B}{6}\right)$ is a local maximum and the "only critical number in town test" tells us that a local extremum is global.