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The problem you shared asks to **obtain the maximum of the function** \( f(x, y) = 9 - x^2 - y^2 \), subject to the constraint \( x + y = 3 \).

To solve this, we can use **Lagrange multipliers**. Here's a step-by-step outline of the solution:

### 1. Define the Lagrangian function:
The Lagrangian is defined as:
\[
\mathcal{L}(x, y, \lambda) = f(x, y) + \lambda (g(x, y))
\]
where \( g(x, y) = x + y - 3 \) is the constraint equation, and \( \lambda \) is the Lagrange multiplier.

Thus, the Lagrangian function becomes:
\[
\mathcal{L}(x, y, \lambda) = 9 - x^2 - y^2 + \lambda (x + y - 3)
\]

### 2. Take partial derivatives:
We now take partial derivatives of the Lagrangian with respect to \( x \), \( y \), and \( \lambda \) and set them equal to zero.

\[
\frac{\partial \mathcal{L}}{\partial x} = -2x + \lambda = 0 \quad \Rightarrow \quad \lambda = 2x
\]
\[
\frac{\partial \mathcal{L}}{\partial y} = -2y + \lambda = 0 \quad \Rightarrow \quad \lambda = 2y
\]
\[
\frac{\partial \mathcal{L}}{\partial \lambda} = x + y - 3 = 0 \quad \Rightarrow \quad x + y = 3
\]

### 3. Solve the system of equations:
From the first two equations, we get \( \lambda = 2x \) and \( \lambda = 2y \), which implies:
\[
2x = 2y \quad \Rightarrow \quad x = y
\]

Substitute this into the constraint \( x + y = 3 \):
\[
x + x = 3 \quad \Rightarrow \quad 2x = 3 \quad \Rightarrow \quad x = \frac{3}{2}
\]

Since \( x = y \), we also have \( y = \frac{3}{2} \).

### 4. Find the maximum value:
Now substitute \( x = \frac{3}{2} \) and \( y = \frac{3}{2} \) into the original function \( f(x, y) \):
\[
f\left( \frac{3}{2}, \frac{3}{2} \right) = 9 - \left( \frac{3}{2} \right)^2 - \left( \frac{3}{2} \right)^2 = 9 - \frac{9}{4} - \frac{9}{4} = 9 - \frac{18}{4} = 9 - 4.5 = 4.5
\]

### Conclusion:
The maximum value of \( f(x, y) \), subject to the constraint \( x + y = 3 \), is **4.5**, and it occurs at \( x = y = \frac{3}{2} \).

Would you like further details, or do you have any questions?

Here are five related questions to deepen understanding:
1. How do Lagrange multipliers work in optimization problems with multiple constraints?
2. What is the geometric interpretation of the solution in this problem?
3. Can the Hessian matrix be used here to confirm if the solution is a maximum?
4. How would the solution change if the constraint were \( x + y = 5 \) instead of \( 3 \)?
5. How would you find the minimum value of the same function under a different constraint?

**Tip:** Always verify whether you're solving for a maximum or minimum by checking the second-order conditions using the Hessian matrix or analyzing the function behavior!

Obtain the maximum of f(x, y) = 9 - x^2 - y^2, subject to x + y = 3.

Learn how to use the method of Lagrange multipliers to find the maximum of the function f(x, y) = 9 - x^2 - y^2, subject to the constraint x + y = 3. This step-by-step solution covers key concepts in multivariable calculus and optimization, suitable for undergraduate-level calculus students.

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