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suppose c(x) = x^3-10x^2-30x, where x is measured in thousands of units. is there production level that minimizes average cost?

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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

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