Let's analyze and solve the given problem step by step. The average cost function is:

$C(x) = 4x^2 - 100x + 800$

where $x$ is the production in thousands of monitors per month.

(i) Average cost at 5,000 monitors per month

Since $x$ represents production in thousands, $x = 5$ for 5,000 monitors. Substitute $x = 5$ into $C(x)$:

$C(5) = 4(5^2) - 100(5) + 800$ $C(5) = 4(25) - 500 + 800 = 100 - 500 + 800 = 400$

Answer: The average cost per unit is 400 euros when producing 5,000 monitors per month.

(ii) Production level to minimize average cost

To minimize $C(x)$, find the vertex of the quadratic function. For a quadratic function $ax^2 + bx + c$, the $x$-coordinate of the vertex is given by:

$x = -\frac{b}{2a}$

Here, $a = 4$ and $b = -100$. Substituting:

$x = -\frac{-100}{2(4)} = \frac{100}{8} = 12.5$

Thus, the average cost is minimized when $x = 12.5$, i.e., when 12,500 monitors are produced per month.

Answer: The production level to minimize average cost is 12,500 monitors per month.

(iii) Minimum possible average cost

To find the minimum average cost, substitute $x = 12.5$ into $C(x)$:

$C(12.5) = 4(12.5^2) - 100(12.5) + 800$ $C(12.5) = 4(156.25) - 1250 + 800 = 625 - 1250 + 800 = 175$

Answer: The minimum possible average cost per unit is 175 euros.

Final Summary:

1. Average cost at 5,000 monitors per month: 400 euros.
2. Production level to minimize cost: 12,500 monitors per month.
3. Minimum average cost: 175 euros.

Would you like further clarification or a detailed graph of the function?

Here are 5 additional questions to extend your understanding:

1. What happens to the cost if production is increased beyond 12,500 monitors?
2. How does the coefficient of $x^2$ affect the shape of the graph?
3. What is the significance of the constant term $800$ in the function?
4. How would you calculate the total cost instead of the average cost using this data?
5. How can this function be used for profit analysis if a selling price is given?

Tip: When dealing with quadratic cost functions, always look for the vertex to find the optimal production level or cost.