To find the profit-maximizing output and price for the monopolist, we need to go through several steps involving the demand function, revenue function, cost function, and conditions for profit maximization. Let's break it down step by step.

Step 1: Define Variables and Functions

1. Demand Function: $p = \frac{50}{x}$, where $p$ is the price and $x$ is the quantity demanded.
2. Average Cost (AC) Function: $AC = 0.5 + \frac{1000}{x}$.
3. Total Cost (TC) Function: Since $AC = \frac{TC}{x}$, the total cost can be derived from the average cost: $TC = AC \cdot x = \left(0.5 + \frac{1000}{x}\right) \cdot x = 0.5x + 1000$

Step 2: Revenue Function

The total revenue (TR) is given by price times quantity: $TR = p \cdot x = \frac{50}{x} \cdot x = 50$

Thus, the total revenue is constant and equal to 50, independent of $x$.

The marginal revenue (MR) is the derivative of the total revenue with respect to quantity ($x$). Since $TR = 50$, we have: $MR = \frac{d(TR)}{dx} = 0$

Step 3: Marginal Cost

The marginal cost (MC) is the derivative of the total cost with respect to quantity: $TC = 0.5x + 1000$ $MC = \frac{d(TC)}{dx} = 0.5$

Step 4: Profit Maximization Condition

For profit maximization, the monopolist sets marginal revenue (MR) equal to marginal cost (MC). From our calculations: $MR = 0 \quad \text{and} \quad MC = 0.5$

This shows that marginal revenue is not equal to marginal cost, which contradicts the typical condition for profit maximization in a monopoly. It seems that the total revenue being constant indicates the monopolist has no real ability to affect revenue through changes in quantity. In such a case, other factors such as fixed production costs would determine the profit-maximizing output.

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Would you like to explore any part of this solution in more detail or discuss the implication of these results? Here are some related questions:

1. How does the constant revenue impact the monopolist's pricing strategy?
2. How does the average cost structure affect profit in the long term?
3. Why does the marginal cost not change with production level?
4. Can there be multiple profit-maximizing outputs in such scenarios?
5. How could external market factors influence this monopolist's decisions?

Tip: In some monopoly models, marginal revenue can become zero when demand is perfectly inelastic. In such cases, cost minimization plays a key role in maximizing profits.