If each MP3 player costs the store $70, at what price should the store sell the MP3 players to maximize profit in the following problem?A store...

The store sells 120 MP3 players at $100 each. For every $2
increase in the price, the number of MP3 players sold is reduced by 1. The cost of one
MP3 player is $70.

Let the price at which the MP3 players
are sold to maximize profit be P. The number of MP3 players sold is 120 - [(P -
100)/2]

The profit per MP3 player is P -
70.

Total profit is (P - 70)(120 - (P -
100)/2)

=> 120P - P(P - 100)/2 - 120*70 + 70(P -
100)/2

=> 120P - P^2/2 + 50P - 8400 + 35P -
3500

=> -P^2/2 + 205P -
11900

To maximise the profit take the derivative of -P^2/2
+ 205P - 11900 and solve for P.

-2P/2 + 205 =
0

=>-P + 205 =
0

=> -P =
-205

=> P = 205

But
only a price increase by multiples of 2 changes the number of MP3 players sold by a
whole number. This makes the optimal price either $204 or
$206.

The price to maximize profits is either
$204 or $206