A store has been selling 500 DVD burners sold per week at 300 each. A market survey indicates that for each $ 20 rebate offered to buyers, the number of units sold will increase by 40 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue?

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.