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The task in the image contains two parts that involve cost functions and price adjustments due to competition in the market. Here’s a breakdown of the key components:

### 1. **Understanding the Cost Function \( K(x) \)**

The problem provides two cost functions for a company that is adjusting to competitive market pressures:

- The **initial total cost function** is given by:
  \[
  K(x) = 0.5x^3 - 60x^2 + 2,500x + 40,000
  \]
  This function includes fixed costs (40,000 GE) and variable costs dependent on the quantity \( x \).

- A new **adjusted total cost function** is introduced, depending on an optimization parameter \( c \) (natural number):
  \[
  K_c(x) = 0.5x^3 - 60x^2 + 600cx + 40,000
  \]
  Here, \( c \) affects the linear term, representing some production adjustment that impacts variable costs.

### 2. **Part A: Find \( c \) to Meet Price of 800 GE/ME**

The problem asks to find the value of \( c \) so that the company can offer its product at a price of 800 GE/ME.

We must equate the **marginal cost** (the derivative of the cost function \( K_c(x) \)) to 800 GE/ME and solve for \( c \). The marginal cost is the derivative of \( K_c(x) \), denoted \( K'_c(x) \).

### 3. **Part B: Determine the Fixed Costs \( K_f \)**

The second part asks for the **maximum share of fixed costs** \( K_f \) that can be covered at a price of 800 GE/ME in the short term. We need to use the cost structure and price assumptions to calculate this.

Let me now compute both parts step by step.The solution for \( c \), the optimization parameter, depends on the quantity \( x \) produced. Specifically, the expression for \( c \) is:

\[
c = -0.0025x^2 + 0.2x + \frac{4}{3}
\]

This means that the value of \( c \) is not a constant, but rather a function of the production quantity \( x \). 

Next, I'll calculate the maximum fixed costs \( K_f \) that can be covered at a price of 800 GE/ME. This involves analyzing the cost structure at that price level, and determining how much of the fixed cost (40,000 GE) is covered. I'll proceed with this calculation now.The solution for the production quantity \( x \) that allows the company to cover its average costs at a price of 800 GE/ME includes one real and two complex values:

- The real solution is approximately \( x = -14.82 \), which is not physically meaningful since production quantity cannot be negative.

This suggests that no feasible production quantity \( x \) can ensure the average cost equals 800 GE/ME under the given cost structure. The company would not be able to cover its costs at this price level in the short term, using the original cost function.

If you'd like, I can explore alternative approaches to this calculation or clarify further details.

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Would you like more details on this analysis, or should I help you with a different aspect? Here are 5 follow-up questions you might want to explore:

1. What happens if we adjust the cost function further to find a feasible \( x \)?
2. How would lowering fixed costs impact the result?
3. Could a different value of \( c \) lead to feasible production quantities?
4. What are the implications of a negative production quantity in economic terms?
5. How does the marginal cost function behave for different price levels?

**Tip**: In optimization problems, checking both marginal and average cost functions helps assess price feasibility for long-term sustainability.

Determine the value of c for the cost function and the maximum share of fixed costs at a price of 800 GE/ME.

This task involves optimizing a cost function to determine the parameter c and calculating the maximum share of fixed costs covered at a price of 800 GE/ME. Suitable for university students studying economics and mathematics.

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