The problem you've provided is written in Japanese. Here's a translation and breakdown:

Problem 1. (Selection)

Given real numbers $x$ and $y$ that satisfy the equation $x^2 + 3y^2 = 9$, find the maximum and minimum values of $x + y^2 - 1$. Additionally, find the values of $x$ and $y$ when the maximum and minimum values are achieved.

To solve this, we will approach it step by step:

Step 1: Set up the objective function

We are tasked with maximizing and minimizing the expression $f(x, y) = x + y^2 - 1$.

Step 2: Given constraint

The constraint is $g(x, y) = x^2 + 3y^2 - 9 = 0$, which defines an ellipse.

Step 3: Use Lagrange multipliers

We can use the method of Lagrange multipliers to solve this constrained optimization problem. This involves solving the system:

$\nabla f(x, y) = \lambda \nabla g(x, y)$

where $\lambda$ is the Lagrange multiplier.

1. $\nabla f(x, y) = (1, 2y)$
2. $\nabla g(x, y) = (2x, 6y)$

Step 4: Solve the system of equations

This gives us the following system:

$1 = \lambda (2x)$ $2y = \lambda (6y)$

Additionally, we have the constraint equation $x^2 + 3y^2 = 9$.

Let me solve these equations.The solutions to the system are:

1. $(x, y) = (-3, 0)$
2. $(x, y) = \left( \frac{3}{2}, -\frac{3}{2} \right)$
3. $(x, y) = \left( \frac{3}{2}, \frac{3}{2} \right)$
4. $(x, y) = (3, 0)$

Step 5: Calculate the values of $f(x, y) = x + y^2 - 1$

We now substitute these values into the function $f(x, y) = x + y^2 - 1$ to find the maximum and minimum values. Let's compute this next.The values of the function $f(x, y) = x + y^2 - 1$ at the critical points are:

1. $f(-3, 0) = -4$
2. $f\left( \frac{3}{2}, -\frac{3}{2} \right) = \frac{11}{4}$
3. $f\left( \frac{3}{2}, \frac{3}{2} \right) = \frac{11}{4}$
4. $f(3, 0) = 2$

Conclusion:

• The minimum value is $-4$ at $(x, y) = (-3, 0)$.
• The maximum value is $\frac{11}{4}$ at $\left( \frac{3}{2}, \pm \frac{3}{2} \right)$.

Would you like more details on any part of the solution or have any additional questions?

Here are five related questions to consider:

1. How does the method of Lagrange multipliers work for constrained optimization problems?
2. What would happen if the constraint was a circle instead of an ellipse?
3. Could the problem be solved using another method, such as substitution?
4. How does the shape of the ellipse influence the values of $x$ and $y$?
5. How are maxima and minima determined geometrically in such optimization problems?

Tip: Always check that the critical points satisfy both the objective function and the constraint to avoid calculation errors.