Answer:
The demand function is
p(x) = 550 – (x/2)
The revenue function is
R(x) = 550x – (x²/2)
In order to maximize revenue, the store should offer a rebate of $25.
Explanation:
If x is the number of DVD burners sold per week, then the weekly increase in sales is x – 500.
For each increase of 40 units sold, the price is decreased by $20.
So for each additional unit sold, the decrease in price will be (1/40)*20 and the demand function is
p(x) = 300 – (20/40)*(x – 500) = 550 – (x/2)
The revenue function is
R(x) = x*p(x) = 550x – (x²/2)
Since R'(x) = 550 – x, we see that R'(x) = 0 when x = 550.
This value of x gives an absolute maximum by the First Derivative Test (or simply by observing that the graph of R is a parabola that opens downward).
The corresponding price is
p(550) = 550 – (550/2) = 275
and the rebate is 300 – 275 = 25.
Therefore, to maximize revenue, the store should offer a rebate of $25.
