{Use of Tech} A different interpretation of marginal cost Suppose a large company makes 25,000 gadgets per year in batches of x items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, planners have determined that the total cost C(x) of producing 25,000 gadgets in batches of x items at a time is given by C(x) = 1,250,000+125,000,000 / x + 1.5x.
a. Determine the marginal cost and average cost functions. Graph and interpret these functions.

Verified step by step guidance

1. Identify the total cost function C(x) = 1,250,000 + 125,000,000 / x + 1.5x, which represents the cost of producing 25,000 gadgets in batches of x items.
2. To find the marginal cost function, calculate the derivative of the total cost function C(x) with respect to x, which is C'(x). This will give you the rate of change of the total cost as the batch size changes.
3. Next, determine the average cost function A(x) by dividing the total cost C(x) by the number of batches produced, which is 25,000 / x. This gives A(x) = C(x) / (25,000 / x).
4. Graph both the marginal cost function C'(x) and the average cost function A(x) on the same set of axes to visually compare their behaviors as x varies.
5. Interpret the graphs by analyzing where the marginal cost is less than, equal to, or greater than the average cost, and discuss the implications for production efficiency and cost management.