We are given the cost function:

$C(x) = x^3 - 10x^2 + 34x$

Step 1: Find the production level where the average cost is minimized.

The average cost function $AC(x)$ is obtained by dividing the total cost $C(x)$ by the number of units produced $x$:

$AC(x) = \frac{C(x)}{x} = \frac{x^3 - 10x^2 + 34x}{x} = x^2 - 10x + 34$

To minimize the average cost, we take the derivative of $AC(x)$ and set it equal to zero to find critical points:

$AC'(x) = \frac{d}{dx}(x^2 - 10x + 34) = 2x - 10$

Setting $AC'(x) = 0$:

$2x - 10 = 0$

Solving for $x$:

$x = 5$

Thus, the production level at which the average cost is minimized is $x = 5$.

Step 2: Verify that at this production level, the marginal cost and average cost are equal.

The marginal cost $MC(x)$ is the derivative of the total cost function $C(x)$:

$MC(x) = \frac{d}{dx}(x^3 - 10x^2 + 34x) = 3x^2 - 20x + 34$

Substituting $x = 5$ into $MC(x)$:

$MC(5) = 3(5)^2 - 20(5) + 34 = 3(25) - 100 + 34 = 75 - 100 + 34 = 9$

Now, substitute $x = 5$ into the average cost function $AC(x)$:

$AC(5) = (5)^2 - 10(5) + 34 = 25 - 50 + 34 = 9$

Therefore, at $x = 5$, both the marginal cost and the average cost are equal to $9$.

Step 3: Graph the average cost function and verify the result.

To verify, we can graph the average cost function $AC(x) = x^2 - 10x + 34$. At $x = 5$, the minimum point should occur, and we have already established that $AC(5) = 9$, matching the marginal cost at that production level.

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Would you like more details or help with the graph?

Here are 5 relative questions that expand on this:

1. How does the shape of the graph change with variations in the coefficients of the cost function?
2. What is the economic significance of the marginal cost being equal to the average cost?
3. How would the solution change if we added a constant to the cost function?
4. Can you explain the relationship between fixed costs and the average cost function?
5. How do we interpret the second derivative of the average cost function in terms of concavity?

Tip: Always remember that the production level where average cost is minimized is the point where marginal cost equals average cost.