Business and Economics
62. Production level Suppose that c(x)=x^3-20x^2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

Verified step by step guidance

1. First, understand that the average cost function is given by dividing the total cost function c(x) by the number of items x. So, the average cost function A(x) is A(x) = c(x)/x = (x^3 - 20x^2 + 20000x)/x.
2. Simplify the average cost function A(x) by dividing each term in the numerator by x. This results in A(x) = x^2 - 20x + 20000.
3. To find the production level that minimizes the average cost, take the derivative of the average cost function A(x) with respect to x. This gives A'(x) = 2x - 20.
4. Set the derivative A'(x) equal to zero to find the critical points. Solve the equation 2x - 20 = 0 to find the value of x that minimizes the average cost.
5. Verify that the critical point found is indeed a minimum by using the second derivative test. Compute the second derivative A''(x) = 2, which is positive, indicating that the function is concave up at the critical point, confirming a minimum.

Key Concepts

Average Cost Function

The average cost function is derived by dividing the total cost function by the number of items produced, x. It represents the cost per unit of production and is crucial for determining the production level that minimizes costs. In this problem, the average cost function is c(x)/x, which simplifies to x^2 - 20x + 20,000.

Critical Points and Optimization

To find the production level that minimizes the average cost, we need to identify critical points by taking the derivative of the average cost function and setting it to zero. This process helps locate points where the function's slope is zero, indicating potential minima or maxima. Solving for these points allows us to determine the optimal production level.

Second Derivative Test

The second derivative test is used to confirm whether a critical point is a minimum or maximum. By evaluating the second derivative at the critical points, we can determine the concavity of the function. If the second derivative is positive, the function is concave up, indicating a local minimum, which is essential for ensuring the production level minimizes average costs.