Question

A company has found that the relationship between the price p and the demand z for a particular product is given approximately by p=1156 -0.13. The company also knows that the cost of producing the product is given by C(x) = 820 + 394.. Find P(x), the profit function P(z) - Now use the profit function to do the following A) Find the average of the values of all local maxima of P Vote if there are no local maxima,

Answer 1

The profit function is : P(x)= −
`0.13^(3)`+ 762x −820 and x = 40.512

Profit Function

P(x)=x⋅p−C(x)

P(x)=x⋅(1156−
`0.13^(2)`)−(820+394x)

P(x)=1156x−
`0.13^(3)`−820 - 394x)

P(x)= −
`0.13^(3)`+ 762x −820

The profit function is: P(x)= −
`0.13^(3)`+ 762x −820

Finding Local Maxima

Find the critical points by setting the derivative of P(x) = 0

P’(x) = 0

`(d)/(dx)-0.13^(3)` +
`(d)/(dx)762x` - ddx(820) = 0

`-0.39x^(2)`+ 762 = 0

`x^(2)` =
`(762)/(0.39)`

x =
`\sqrt{(762)/(0.39) }`

x = 40.512

Calculating the second derivative of P(x),

`(d)/(dx)-0.39x^(2)` +
`(d)/(dx)762` = 0

-0.78x= 0

x = 0

Therefore, the company can maximise its profit by producing and selling the product at approximately 40.512 units

The profit function is : P(x)= −0.13x3+ 762x −820 and x = 40.512

Question

A company has found that the relationship between the price p and the demand z for a particular product is given approximately by p =1156 -0.13x2. The company also knows that the cost of producing the product is given by C(x) = 820 + 394x. Find P(x), the profit function. Now use the profit function to do the following

A) Find the average of the x values of all local maxima of P

Note if there are no local maxima, enter-1000