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?

1 Answer
Jim H
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 (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.

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