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Let's analyze each of the statements provided based on the demand curve and average total cost (ATC) function given:

1. **Demand Curve**: \( Q = 80 - \frac{4}{3} P \)
2. **Average Total Cost (ATC)**: \( ATC = 0.5Q + 10 + \frac{1000}{Q} \)

### To find the not true statement:

#### A. The maximum total profit is positive
To verify this, we need to compute the profit function:
\[ \text{Profit} = \text{Total Revenue} - \text{Total Cost} \]
Where:
\[ \text{Total Revenue (TR)} = P \times Q \]
\[ \text{Total Cost (TC)} = ATC \times Q \]
First, solve for \( P \) in terms of \( Q \) from the demand curve:
\[ P = 60 - \frac{3}{4}Q \]
Now, calculate TR:
\[ TR = Q(60 - \frac{3}{4}Q) = 60Q - \frac{3}{4}Q^2 \]
Calculate TC:
\[ TC = Q(ATC) = Q(0.5Q + 10 + \frac{1000}{Q}) = 0.5Q^2 + 10Q + 1000 \]
Thus, the profit equation becomes:
\[ \text{Profit} = 60Q - \frac{3}{4}Q^2 - (0.5Q^2 + 10Q + 1000) \]
\[ = 50Q - 1.25Q^2 - 1000 \]
To determine if the maximum total profit is positive, we need to find the maximum of this quadratic function.

#### B. The maximum value of the total revenue function is 1,200
The total revenue function TR \( = 60Q - \frac{3}{4}Q^2 \). The maximum of a quadratic function \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \):
\[ Q_{\text{max}} = -\frac{60}{-1.5} = 40 \]
\[ TR_{\text{max}} = 60(40) - \frac{3}{4}(40)^2 = 2400 - 1200 = 1200 \]
This statement is true.

#### C. The profit maximizing output level is \( Q = 20 \)
To find this, we would maximize the profit equation calculated above. Setting the derivative equal to zero gives us the profit-maximizing output.

#### D. The most profitable price is \( P = 45 \)
We check this by inserting \( Q = 20 \) (if true for statement C) into the price equation derived from the demand curve.

#### E. The total cost of production is \( TC = 0.5Q^2 + 10Q + 1,000 \)
This is directly taken from the ATC function, and since \( TC = ATC \times Q \), the simplification shows that this statement is true.

### Calculation:
We need to calculate the profit function, find the vertex for maximum revenue, and verify profit maximization output and the corresponding price. Let's do these calculations.Based on the calculations:

- The profit maximizing output level is \( Q = 20 \), which is true.
- The corresponding price that maximizes profit is \( P = 45 \), which is also true.
- The maximum profit at \( Q = 20 \) is \(-500\), indicating a loss, not a profit.

### Analyzing the Statements:
- **A. The maximum total profit is positive** — This is false. The calculation shows a maximum profit of \(-500\), indicating a loss.
- **B. The maximum value of the total revenue function is 1,200** — This is true based on earlier analysis.
- **C. The profit maximizing output level is \( Q = 20 \)** — This is true.
- **D. The most profitable price is \( P = 45 \)** — This is true.
- **E. The total cost of production is \( TC = 0.5Q^2 + 10Q + 1,000 \)** — This is true.

Thus, the statement that is **not true** is **A**: "The maximum total profit is positive."

If you need further details or have more questions on this topic, please let me know! Here are some additional questions you might find interesting:
1. How does changing the fixed cost affect the profit-maximizing output and profit?
2. What happens to the break-even output level if the selling price per unit increases?
3. How would introducing government subsidies affect the profit curve?
4. What is the impact of variable cost changes on total revenue and profit?
5. Explore how economies of scale influence average total costs.

**Tip**: When analyzing economic functions, always check the roots and critical points of the equations to understand maximum and minimum values which are essential for strategic decision-making.

The demand curve facing a firm is Q = 80 - (4/3)P and the average total cost of production is ATC = 0.5Q + 10 + 1,000/Q. Which of the following is not true? A. The maximum total profit is positive B. The maximum value of the total revenue function is 1,200 C. The profit maximizing output level is Q = 20 D. The most profitable price is P = 45 E. The total cost of production is TC = 0.5Q^2 + 10Q + 1,000

Explore the profit maximization process through analyzing a firm's demand curve and average total cost function. This detailed explanation includes solving for revenue, cost, and profit functions, and determining the correctness of given economic statements.

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