1. (30 points) 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 this expression, we get:
A(x) = (10000/x) + 5 + (1/9)x
b. To minimize the average cost, we need to find the minimum value of the average cost function A(x). We can do this by finding the critical points of A(x).
To find the critical points, we need to find the derivative of A(x) with respect to x, and set it equal to zero:
A'(x) = -10000/x^2 + 1/9
Setting A'(x) equal to zero and solving for x, we get:
0 = -10000/x^2 + 1/9
Multiplying both sides of the equation by x^2:
0 = -10000 + x^2/9
Rearranging the equation:
x^2/9 = 10000
Taking the square root of both sides:
x/3 = 100
x = 300
Therefore, the manufacturer should produce 300 items in order to minimize the 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 this expression, we get:
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 need to find the derivative of the total cost function c(x) with respect to x:
C'(x) = d/dx (10000 + 5x + (1/9)x^2)
C'(x) = 5 + (2/9)x
Therefore, C'(x) = 5 + (2/9)x.
e. The total cost function C(x) has a critical point when the derivative C'(x) is equal to zero. From part d, we found that C'(x) = 5 + (2/9)x.
Setting C'(x) equal to zero and solving for x, we get:
0 = 5 + (2/9)x
Multiplying both sides of the equation by 9:
0 = 45 + 2x
Rearranging the equation:
2x = -45
x = -45/2
Since the number of items produced cannot be negative, there is no critical point for C(x).
Therefore, the average cost when C(x) has a critical point is undefined.