We are given the cost function for manufacturing $x$ items as:

$C(x) = 3x^3 - 36x^2 + 13,000x$

To find the production level that will minimize the average cost, we first need to compute the average cost function, which is given by:

$AC(x) = \frac{C(x)}{x}$

Step 1: Calculate the Average Cost Function

The average cost function is:

$AC(x) = \frac{3x^3 - 36x^2 + 13,000x}{x}$

Simplifying:

$AC(x) = 3x^2 - 36x + 13,000$

Step 2: Minimize the Average Cost Function

To find the production level that minimizes the average cost, we need to differentiate $AC(x)$ with respect to $x$ and set the derivative equal to zero.

$AC'(x) = \frac{d}{dx}\left(3x^2 - 36x + 13,000\right)$

$AC'(x) = 6x - 36$

Step 3: Solve for Critical Points

Set $AC'(x) = 0$ to find the critical points:

$6x - 36 = 0$

$6x = 36$

$x = 6$

Step 4: Confirm it's a Minimum

To confirm that $x = 6$ is a minimum, we check the second derivative of $AC(x)$:

$AC''(x) = 6$

Since $AC''(x) = 6 > 0$, the function is concave up at $x = 6$, which confirms that this is a minimum.

Conclusion

The production level that minimizes the average cost is 6 items.

---

Do you want any additional details or have questions about this process? Here are five follow-up questions you might consider:

1. What is the minimum average cost when producing 6 items?
2. How does the total cost change as production increases?
3. Can the production level be fractional, or must it always be a whole number?
4. How does the second derivative test help confirm the nature of critical points?
5. What happens to the average cost at very large production levels?

Tip: Always verify if the critical points found are maximum or minimum by checking the second derivative.