Question

Hercules Films is deciding on the price of the video release of its film Son of Frankenstein. Its marketing people estimate that at a price of p dollars, it can sell a total of q = 210,000 − 15,000p copies. (a) What is the revenue function? R(p)= (b) What price will bring in the greatest revenue? p = dollars Second derivative test: Your answer above is a critical point for the revenue function. To show it is a maximum, calculate the second derivative of the revenue function. R"(p)= Evaluate R"(p) at your critical point. The result is , which means that the revenue is at the critical point, and the critical point is a maximum.

Answer 1

Answer:

Revenue =  - 15,000p² + 210,000p

Vertex at price = 7

Revenue = 735,000

Explanation:

Revenue = price x quantity

R(p) = (210,000 - 15,000p) = 210,000p - 15,000p²

We calcualte the vertex:

-b/2a = -(-210,000) / (15,000 x 2 ) = 210,000 / 30,000 = 7

-15,000 x 49 + 210,000 x 7 = 735000

R(p) =  - 15,000p² + 210,000p

We derivate using the following identity:

$ax^{b} = bax^{b-1}$

R(p)' =  - 30,000p + 210,000

R(p)'' =  -30,000

As the second derivate is constant negative there is only one critical point and, is a maximum.