Question

A company has found that the relationship between the price p and the demand x for a particular product is given approximately by
p=1269−0.2x2.
The company also knows that the cost of producing the product is given by C(x)=830+398x.

(A) Find P(x), the profit function.
P(x) =

Now use the profit function to do the following:

(B) Find the average of the x values of all local maxima of P.
Note: If there are no local maxima, enter "none".

Average of x values =

Answer 1

We know that the profit from an item is the sale price minus the cost price. Thus,
1269 - 0.2x² - 830 - 398x
P(x) = -0.2x² - 398x + 439
Local maxima are located at the points where the first derivative of a function are 0
P'(x) = -0.4x - 398
398 = -0.4x
x = -995
There is only one maximum so the average value is -995