Question

For the cost function C(x)=3000+48 x+0.003 x³ find:

A) The production level that will minimize the average cost.
B) The minimal average cost.

Answer 1

To find the production level that minimizes the average cost and the minimal average cost for a given cost function, calculate the average cost function, find its derivative, set the derivative to zero to find the minimizing quantity, and then calculate the minimal average cost by substituting back into the average cost function.

Step-by-step explanation:

The student asked about finding the production level that minimizes the average cost and also what the minimal average cost is given a specific cost function, C(x) = 3000 + 48x + 0.003x³. To tackle this question, first, we need to find the average cost function, AC(x), by dividing the cost function by x. Then, to minimize AC(x), we take the derivative and set it to zero to solve for x, which gives us the quantity that minimizes the average cost. The minimal average cost is then calculated by substituting this x value back into the average cost function.

Step-by-step:

1. Calculate the average cost (AC) function: AC(x) = C(x) / x
2. Find the derivative of AC(x) with respect to x.
3. Set the derivative equal to zero and solve for x to find the production level that minimizes AC.
4. Substitute this x value into AC(x) to find the minimal average cost.