https://file.aiquickdraw.com/mathbot/file/1726810881678ekj5pntu.jpg

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.

実数 x, y が x^2 + 3y^2 = 9 を満たすとき, x + y^2 - 1 の最大値と最小値を求めなさい。また、最大値と最小値をとるときの x, y の値をそれぞれ求めなさい。

Solve for the maximum and minimum values of the function x + y^2 - 1, subject to the constraint x^2 + 3y^2 = 9. This constrained optimization problem utilizes Lagrange multipliers and is suitable for high school or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation