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The problem you're dealing with involves a production company that operates with a total cost function (\(C(x)\)) which is S-shaped. The fixed costs are 20 GE, and we are given several points to help determine the structure of the cost function:

### Known Points and Characteristics:
1. **Fixed costs** = 20 GE, which implies \(C(0) = 20\).
2. At a production level of 3 ME (million units), the total costs are 56 GE: \(C(3) = 56\).
3. The cost growth at 1 ME is 15 GE, meaning the marginal cost at \(x = 1\) is 15: \(C'(1) = 15\).
4. At the capacity limit of \(x = 8\), the maximum costs are 196 GE: \(C(8) = 196\).
5. The cost function changes from degressive (decreasing growth) to progressive (increasing growth), which suggests that we might be dealing with a cubic function due to its S-shape (common for cost functions that transition from concave to convex behavior).

### Approach:
To find the total cost function \(C(x)\), we can assume a cubic function of the form:

\[
C(x) = ax^3 + bx^2 + cx + d
\]

Where:
- \(d\) represents the fixed costs (20 GE),
- \(a, b, c\) are coefficients we need to solve for based on the given conditions.

We will use the following conditions to set up the system of equations:
1. \(C(0) = 20\),
2. \(C(3) = 56\),
3. \(C'(1) = 15\),
4. \(C(8) = 196\).

Let's calculate the values of \(a\), \(b\), and \(c\). I'll solve this system of equations for you.The total cost function \(C(x)\) based on the given conditions is:

\[
C(x) = x^3 - 9x^2 + 30x + 20
\]

This cubic function reflects the S-shaped behavior of the cost, transitioning from a degressive to a progressive growth pattern. The fixed cost is \(20\) GE, and the curve satisfies the points provided.

Would you like a deeper explanation of any part of the solution or further clarifications?

Here are 5 related questions to consider:
1. How can we interpret the progressive and degressive regions of the cost function?
2. What is the significance of marginal cost in this context?
3. How does this cubic cost function compare to a linear cost function?
4. What would happen if the capacity limit changed to a different value?
5. How can this cost function be used for decision-making in production planning?

**Tip:** When analyzing cost functions, the second derivative gives insights into the rate of change of marginal cost (helpful to identify inflection points).

Ein Produktionsunternehmen arbeitet mit einer s-förmigen Gesamtkostenfunktion. Die Fixkosten betragen 20 GE. Bei einer Produktionsmenge von 3 ME entstehen Kosten in Höhe von 56 GE. Gleichzeitig wechselt der Gesamtkostenverlauf bei dieser Produktionsmenge von einem degressiven in einen progressiven Verlauf. Bei einer Produktionsmenge von 1 ME beträgt der Kostenzuwachs 15 GE. Die maximalen Kosten des Betriebes an der Kapazitätsgrenze x = 8 betragen 196 GE. Wie lautet die Gleichung der Gesamtkostenfunktion?

Solve for the cubic total cost function of a production company with fixed costs of 20 GE. Analyze the S-shaped cost behavior, degressive to progressive cost shifts, and find the total cost function at maximum capacity.

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