To find the extrema of the function $f(x, y) = 2x^2 + 3y^2$ subject to the constraint $g(x, y) = x^2 + y^2 - 1 = 0$ using Lagrange multipliers, we proceed as follows:

1. Set up the equations: We want to find the points where the gradients of $f$ and $g$ are proportional: $\nabla f = \lambda \nabla g$

   Calculating the gradients:

   • $\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (4x, 6y)$
   • $\nabla g = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y} \right) = (2x, 2y)$

2. Set up the system of equations: From $\nabla f = \lambda \nabla g$, we have: $4x = \lambda (2x)$ $6y = \lambda (2y)$

   Additionally, we have the constraint: $x^2 + y^2 = 1$

3. Analyze the equations: From the first equation, if $x \neq 0$: $\lambda = \frac{4}{2} = 2$

   From the second equation, if $y \neq 0$: $\lambda = \frac{6}{2} = 3$

   Since these two values of $\lambda$ are not equal, we consider the cases where either $x = 0$ or $y = 0$.

4. Case 1: $x = 0$: Substituting into the constraint: $0^2 + y^2 = 1 \implies y = \pm 1$ Then, $f(0, 1) = 2(0)^2 + 3(1)^2 = 3$ $f(0, -1) = 2(0)^2 + 3(-1)^2 = 3$

5. Case 2: $y = 0$: Substituting into the constraint: $x^2 + 0^2 = 1 \implies x = \pm 1$ Then, $f(1, 0) = 2(1)^2 + 3(0)^2 = 2$ $f(-1, 0) = 2(-1)^2 + 3(0)^2 = 2$

6. Case 3: $x \neq 0$ and $y \neq 0$: Setting $2 = \lambda$ and $3 = \lambda$ gives a contradiction, so we ignore this case.

7. Summary of critical points:

   • $f(0, 1) = 3$
   • $f(0, -1) = 3$
   • $f(1, 0) = 2$
   • $f(-1, 0) = 2$

8. Conclusion:

   • Maximum value: $3$ at points $(0, 1)$ and $(0, -1)$
   • Minimum value: $2$ at points $(1, 0)$ and $(-1, 0)$

Would you like further details on any specific step, or do you have any questions? Here are some follow-up questions you might consider:

1. How does the method of Lagrange multipliers apply to other types of constraints?
2. Can you derive the gradients of the functions in a different context?
3. What are some real-world applications of optimization with constraints?
4. How would the solution change if the constraint were different?
5. What is the geometric interpretation of the constraint in this problem?

Tip: Always visualize the constraint and the function to better understand the optimization problem.