Answer:
To find out whether there is a production level that minimizes the average cost, we need to find the average cost function first. The average cost (AC) function is given by:
AC(x) = C(x)/x
where C(x) is the cost function and x is the production level.
Substituting the given cost function into the above equation, we get:
AC(x) = (x^3 - 10x^2 - 30x)/x
AC(x) = x^2 - 10x - 30
To find the production level that minimizes the average cost, we need to find the derivative of the average cost function and set it equal to zero:
AC'(x) = 2x - 10
Setting AC'(x) = 0, we get:
2x - 10 = 0
2x = 10
x = 5
Therefore, the critical point of the average cost function is x = 5, which means that there may be a production level that minimizes the average cost. To confirm whether this is a minimum or maximum point, we need to find the second derivative of the average cost function:
AC''(x) = 2
Since AC''(5) = 2 is positive, we can conclude that the critical point x = 5 is a local minimum of the average cost function. Therefore, there is a production level of 5,000 units that minimizes the average cost