The cost of producing x units of a product is given by
C(x)=800+120x−120ln(x),x≥1.
Find the minimum average cost, which is defined as C(x)/x (the total cost divided by the number of units).
Minimum Average Cost =
To find the minimum average cost, we need to find the value of x that minimizes the expression C(x) / x.
1. Start by differentiating C(x) with respect to x to find the derivative:
C'(x) = 120 - 120 / x
2. Set the derivative equal to zero and solve for x to find the critical points:
120 - 120 / x = 0
Dividing both sides by 120:
1 - 1 / x = 0
Multiplying both sides by x:
x - 1 = 0
x = 1
So, we have found a critical point at x = 1.
3. To determine whether this critical point is a minimum or maximum, we can use the second derivative test. Find the second derivative of C(x) with respect to x:
C''(x) = 120 / x^2
4. Substitute the critical point x = 1 into the second derivative:
C''(1) = 120 / (1^2) = 120
Since the second derivative is positive at x = 1, we can conclude that the critical point is a minimum.
5. Plug the minimum value back into the original equation C(x) to find the minimum average cost:
C(1) = 800 + 120(1) - 120ln(1) = 800 + 120 - 120(0) = 920
Therefore, the minimum average cost is 920.