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The image you've shared contains exercises about costs related to production in the context of an enterprise producing perfume. It covers concepts such as total cost, revenue, profit, average cost, and marginal cost. Here’s a breakdown of what’s presented:

1. **Cost Function:**
   \[
   C(x) = 0.5x^2 + 2x + 200
   \]
   This represents the total cost in hundreds of euros for producing \( x \) liters of perfume. The exercise focuses on \( x \) between 10 and 50 liters of perfume.

2. **Revenue and Profit:**
   The revenue is modeled by the function:
   \[
   R(x) = 25x
   \]
   This implies that each liter of perfume is sold for 2,500 euros (since 25 * 100 = 2500). 

   The profit \( B(x) \) is given by the difference between the revenue and the cost:
   \[
   B(x) = R(x) - C(x)
   \]
   The goal is to express the profit in terms of \( x \) and determine the maximum profit by finding the optimal quantity of perfume to produce.

3. **Average Cost:**
   The average cost \( C_M(x) \) is the total cost divided by the number of liters produced:
   \[
   C_M(x) = \frac{C(x)}{x}
   \]
   The exercise asks to analyze the behavior of this function and determine where the average cost is minimized.

4. **Marginal Cost:**
   The marginal cost represents the additional cost incurred by producing one more liter of perfume. It is modeled as:
   \[
   C_m(x) = C(x + 1) - C(x)
   \]
   Alternatively, the marginal cost can be found by differentiating the total cost function \( C'(x) \), which is discussed in the later part of the exercise.

### Exercise Breakdown:

- **1. a)** Show that the revenue is modeled by \( R(x) = 25x \).
- **1. b)** Express the profit \( B(x) \) as a function of \( x \).
- **1. c)** Determine the quantity of perfume to maximize the profit.
- **2. a)** Find the variations in the average cost \( C_M(x) \).
- **2. b)** Determine the quantity that minimizes the average cost.
- **3. a)** Calculate the marginal cost for producing 20 liters of perfume, specifically the additional cost from 20 liters to 21 liters.
- **3. b)** Compute the derivative \( C'(x) \) at \( x = 20 \) and compare it with the marginal cost.
- **4. a)** Express the marginal cost \( C_m(x) \) and differentiate the cost function to analyze convexity and solve an equation related to the average and marginal costs.

Would you like me to solve a specific part of these exercises or provide further details on a particular section? Let me know!

### Additional Questions:
1. How is the profit function derived and what does it represent?
2. How do you compute the critical points to maximize profit or minimize costs?
3. What is the significance of the marginal cost compared to the average cost?
4. How can the second derivative test help confirm if a point is a maximum or minimum?
5. What are the implications of convexity in the context of the cost function?

#### Tip: 
When optimizing functions, always compute the first derivative to find critical points and use the second derivative to verify their nature (maximum or minimum).

The image contains exercises related to the costs of producing perfume, focusing on total cost, revenue, profit, average cost, and marginal cost.

Explore cost, revenue, and profit optimization in the context of perfume production. Learn how to model and solve for maximum profit, minimum average cost, and marginal costs using calculus. Suitable for high school and early college students.

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