Final answer:
To find the minimum average variable cost, we extract the variable cost from the total cost function and then calculate its derivative. The critical point where the derivative is zero gives the output level at which AVC is minimum.
Step-by-step explanation:
The student asked for the minimum value of the average variable cost (AVC) for a cost function C = q³ - 9q² + 30q + 25, where C represents cost in thousand dollars, and q is the output in Tons (T). To find the AVC, we must first identify the variable cost portion of the total cost, which here is q³ - 9q² + 30q. We then divide this by the quantity of output q to obtain the AVC.
To determine the minimum point of the AVC, calculus can be used to find the critical points by setting the derivative of the AVC with respect to q to zero and then testing these points to determine the minimum. Given that the question seeks the minimum value of AVC, and not where this minimum occurs, an additional step of evaluating the second derivative may not be necessary unless we want to confirm the nature of the critical point.
The average variable cost (AVC) is calculated by dividing the variable cost by the quantity of output. In this case, the cost function is given as C = q³ - 9q² + 30q + 25. To find the minimum value of the AVC, we need to first find the variable cost function.
The variable cost is the cost that varies directly with the quantity of output, excluding fixed costs. In this case, the variable cost function can be calculated by taking the derivative of the cost function with respect to q.
Once we have the variable cost function, we can calculate the AVC by dividing the variable cost by the quantity of output. To find the minimum value of the AVC, we can set its derivative equal to zero and solve for q. This will give us the quantity of output at which the AVC is minimized.