Astore 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.

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