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The problem states the following:

- The total cost for the monopolist is given as \( C = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the output level.
- The demand law (or price function) is \( p = \beta - \alpha x^2 \), where \( p \) is the price, \( \alpha \) and \( \beta \) are constants, and \( x \) is the output.

You are asked to show that the output for maximum net revenue is:

\[
x = \frac{\sqrt{a^2 + 3a(\beta - b)} - a}{3\alpha}
\]

### Steps to Solve:

1. **Revenue Function:**
   The revenue \( R \) is defined as the product of the price \( p \) and the output \( x \):
   \[
   R(x) = p \cdot x = (\beta - \alpha x^2) \cdot x = \beta x - \alpha x^3
   \]

2. **Profit (Net Revenue):**
   The profit function \( \pi(x) \) is the difference between revenue and cost:
   \[
   \pi(x) = R(x) - C(x) = (\beta x - \alpha x^3) - (ax^2 + bx + c)
   \]
   Simplifying:
   \[
   \pi(x) = \beta x - \alpha x^3 - ax^2 - bx - c
   \]

3. **Maximizing Profit:**
   To maximize the profit, take the first derivative of the profit function with respect to \( x \) and set it equal to zero:
   \[
   \frac{d\pi}{dx} = \frac{d}{dx} \left( \beta x - \alpha x^3 - ax^2 - bx - c \right)
   \]
   The derivative is:
   \[
   \frac{d\pi}{dx} = \beta - 3\alpha x^2 - 2ax - b
   \]
   Set this equal to zero:
   \[
   \beta - 3\alpha x^2 - 2ax - b = 0
   \]
   
4. **Solving the Quadratic Equation:**
   This is a quadratic equation in \( x \):
   \[
   3\alpha x^2 + 2ax + (b - \beta) = 0
   \]
   The solution to a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
   \[
   x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}
   \]
   Here, \( A = 3\alpha \), \( B = 2a \), and \( C = b - \beta \). Substituting these into the quadratic formula:
   \[
   x = \frac{-2a \pm \sqrt{(2a)^2 - 4(3\alpha)(b - \beta)}}{2(3\alpha)}
   \]
   Simplifying:
   \[
   x = \frac{-2a \pm \sqrt{4a^2 + 12\alpha(\beta - b)}}{6\alpha}
   \]
   \[
   x = \frac{-a \pm \sqrt{a^2 + 3\alpha(\beta - b)}}{3\alpha}
   \]

5. **Choosing the Positive Root:**
   Since \( x \) represents output, which must be non-negative, we select the positive root:
   \[
   x = \frac{-a + \sqrt{a^2 + 3\alpha(\beta - b)}}{3\alpha}
   \]

This matches the given expression, proving the result.

---

Do you want further clarifications on this solution? Here are some related questions to consider:

1. What is the relationship between total revenue and marginal revenue in this problem?
2. How would the result change if the demand curve was linear instead of quadratic?
3. What role does the constant \( c \) play in the total cost function if it doesn't appear in the final result?
4. How would you compute the second derivative test to confirm the maximum net revenue?
5. Can you describe the economic interpretation of the discriminant \( a^2 + 3\alpha(\beta - b) \)?

**Tip:** Always remember to check whether the second derivative is negative when solving for maximum values!

If a monopolist has a total cost of C = ax^2 + bx + c and if the demand law is p = β - αx^2, show that the output for maximum net revenue is x = (sqrt(a^2 + 3α(β - b)) - a) / (3α).

Explore the detailed solution to a monopolist's profit maximization problem, where the total cost is C = ax^2 + bx + c and the demand law is p = β - αx^2. The solution involves using the revenue function and optimizing the profit function using the quadratic formula. Learn step-by-step how to find the output that maximizes net revenue.

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