Question

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?

Answer 1

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 `(3N-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+3k` is `N-k`. (It is `N` reduced by 1 per `k`.)

The Revenue will be:

`R(k) = (B+3k)(N-k)`

`= BN-Bk+3Nk-3k^2`

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

`R'(k) = -B+3N-6k`

`R'` is never undefined and is `0` at `k=(3N-B)/6`

The second derivative test tells us that `R((3N-B)/6)` is a local maximum and the "only critical number in town test" tells us that a local extremum is global.