To solve this problem, we need to find the extreme values of $f(x, y) = x - 2y$ subject to the constraint $g(x, y) = x^2 + y^2 = 20$ using the method of Lagrange multipliers.

Step-by-Step Solution

1. Set up the Lagrange Function:
   Define the Lagrange function as: $\mathcal{L}(x, y, \lambda) = f(x, y) - \lambda (g(x, y) - 20)$ So, $\mathcal{L}(x, y, \lambda) = x - 2y - \lambda (x^2 + y^2 - 20)$

2. Compute the Partial Derivatives: We need to find the partial derivatives of $\mathcal{L}$ with respect to $x$, $y$, and $\lambda$, and set them equal to zero. $\frac{\partial \mathcal{L}}{\partial x} = 1 - 2\lambda x = 0$ $\frac{\partial \mathcal{L}}{\partial y} = -2 - 2\lambda y = 0$ $\frac{\partial \mathcal{L}}{\partial \lambda} = -(x^2 + y^2 - 20) = 0$

3. Solve the System of Equations:

   • From $\frac{\partial \mathcal{L}}{\partial x} = 0$:
     $1 = 2\lambda x \Rightarrow \lambda = \frac{1}{2x}$
   • From $\frac{\partial \mathcal{L}}{\partial y} = 0$:
     $-2 = 2\lambda y \Rightarrow \lambda = -\frac{1}{y}$
   • Setting the two expressions for $\lambda$ equal to each other: $\frac{1}{2x} = -\frac{1}{y} \Rightarrow y = -2x$
   • Substitute $y = -2x$ into the constraint $x^2 + y^2 = 20$: $x^2 + (-2x)^2 = 20 \Rightarrow x^2 + 4x^2 = 20 \Rightarrow 5x^2 = 20 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2$
   • For $x = 2$, $y = -4$.
     For $x = -2$, $y = 4$.

4. Calculate $f(x, y)$ at the Points:

   • For $(x, y) = (2, -4)$:
     $f(2, -4) = 2 - 2(-4) = 2 + 8 = 10$
   • For $(x, y) = (-2, 4)$:
     $f(-2, 4) = -2 - 2(4) = -2 - 8 = -10$

5. Identify the Maximum and Minimum Values:

   • The maximum value is $10$.
   • The minimum value is $-10$.

Final Answers:

• Maximum: $\mathbf{10}$
• Minimum: $\mathbf{-10}$

Would you like a more detailed explanation, or do you have any questions?

Here are five additional questions related to this problem:

1. How do Lagrange multipliers help in finding extreme values with constraints?
2. What other methods could you use to solve constrained optimization problems?
3. Why do we set $\nabla f = \lambda \nabla g$ in the method of Lagrange multipliers?
4. Can we have multiple values for $\lambda$ in a solution?
5. What is the significance of the constraint $x^2 + y^2 = 20$?

Tip: When solving constrained optimization problems, always verify that the points satisfy both the constraint and the original function to ensure correct extreme values.