​Suppose that 

c left parenthesis x right parenthesis equals 3 x cubed minus 36 x squared plus 13 comma 000 xc(x)=3x3−36x2+13,000x 

is the cost of manufacturing x items.  Find a production level that will minimize the average cost of making x items.

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.

Suppose that C(x) = 3x^3 - 36x^2 + 13,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

This problem explores how to minimize the average cost of manufacturing x items using calculus-based optimization. Learn how to derive the average cost function from a given cost function, find critical points, and apply the second derivative test. Suitable for high school students in Grades 11-12.

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