Consider a manufacturer whose total cost of producing x items is given by c(x)=10000+5x+1/9x^2.
a. What is the average cost function A(x)= c(x)/x?
b. How many items should the manufacturer produce in order to minimize average cost?
c. What is the smallest average cost?
d. Find C’(x).
e. When does C(x) have a critical point? What is the average cost when C(x) has a critical point?
a. To find the average cost function A(x), divide the total cost function c(x) by the number of items x.
A(x) = c(x)/x = (10000 + 5x + (1/9)x^2) / x
Simplifying the expression further:
A(x) = 10000/x + 5 + (1/9)x
b. To minimize the average cost, we need to find the value of x that minimizes the average cost function A(x). We can do this by finding the critical points of A(x).
To find the critical points, we differentiate A(x) with respect to x and set the derivative equal to zero:
A'(x) = -10000/x^2 + (1/9) = 0
Multiplying through by x^2:
-10000 + (x^2/9) = 0
Simplifying further:
x^2 = 90000
Taking the square root of both sides:
x = ± 300
Since the number of items cannot be negative, we ignore the negative solution. Therefore, the manufacturer should produce 300 items in order to minimize average cost.
c. To find the smallest average cost, substitute the value of x=300 into the average cost function A(x):
A(300) = 10000/300 + 5 + (1/9)(300)
Simplifying the expression:
A(300) = 33.33 + 5 + 33.33
A(300) = 71.66
Therefore, the smallest average cost is 71.66.
d. To find C'(x), we differentiate the total cost function c(x) with respect to x:
C'(x) = 5 + (2/9)x
e. C(x) has a critical point when C'(x) = 0. Setting the derivative equal to zero and solving for x:
5 + (2/9)x = 0
Multiplying through by 9:
45 + 2x = 0
2x = -45
x = -22.5
However, since we're dealing with the number of items produced, which cannot be negative, we disregard this solution.
When C(x) has a critical point, the average cost A(x) can be calculated by substituting the value of the critical point into A(x).