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The problem you provided asks to calculate the maximum and minimum of the function \( A x + B y + C z \), where \( A, B, C \) are positive constants, under the constraints:
- \( x, y, z \geq 0 \)
- \( x y z = 1 \)

It also asks for the generalization of this problem to more than three variables.

### Approach:
1. **Optimization with Lagrange multipliers**: We will use the method of Lagrange multipliers to find the extrema of the objective function \( f(x, y, z) = A x + B y + C z \), subject to the constraint \( g(x, y, z) = x y z - 1 = 0 \).

2. **Lagrange multiplier formulation**: The Lagrangian function is:
   \[
   \mathcal{L}(x, y, z, \lambda) = A x + B y + C z + \lambda (x y z - 1)
   \]
   To find the critical points, we need to compute the partial derivatives of \( \mathcal{L} \) and set them equal to zero:
   - \( \frac{\partial \mathcal{L}}{\partial x} = A + \lambda y z = 0 \)
   - \( \frac{\partial \mathcal{L}}{\partial y} = B + \lambda x z = 0 \)
   - \( \frac{\partial \mathcal{L}}{\partial z} = C + \lambda x y = 0 \)
   - \( \frac{\partial \mathcal{L}}{\partial \lambda} = x y z - 1 = 0 \)

   Solve this system of equations for \( x, y, z \) and \( \lambda \).

3. **Generalization to more than three variables**: The problem can be generalized to \( n \) variables with a similar setup. The function becomes:
   \[
   f(x_1, x_2, \dots, x_n) = A_1 x_1 + A_2 x_2 + \dots + A_n x_n
   \]
   subject to the constraint:
   \[
   x_1 x_2 \dots x_n = 1 \quad \text{and} \quad x_1, x_2, \dots, x_n \geq 0
   \]

Let me know if you'd like to proceed with detailed calculations or a more specific approach for the generalized case. Here are some related questions you might want to explore:

1. How do the Lagrange multipliers work in constrained optimization?
2. How can the solution change if the constants \( A, B, C \) take negative values?
3. What are the applications of this type of optimization in real-world problems?
4. What are the geometric interpretations of the optimization result?
5. How does this problem relate to the Arithmetic-Geometric Mean Inequality?

**Tip**: The method of Lagrange multipliers is a powerful tool for solving optimization problems with constraints. Understanding the theory behind it can help in tackling a wide range of optimization problems.

Siano A, B e C costanti positivi. Calcolare il massimo e il minimo di Ax + By + Cz sotto i vincoli x, y, z ≥ 0 e xyz = 1. Quale sarebbe la generalizzazione a più di 3 variabili?

Learn how to calculate the maximum and minimum of Ax + By + Cz under the constraints x, y, z ≥ 0 and xyz = 1. Explore the generalization to more variables using Lagrange multipliers and understand the link to the Arithmetic-Geometric Mean Inequality.

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