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The problem is asking to find the maximum and minimum values of \( x + y + z \), subject to the constraint:

\[
\frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} = 1.
\]

This can be solved using the **method of Lagrange multipliers**. Here's the general approach:

### Step 1: Set up the Lagrange function
Let \( f(x, y, z) = x + y + z \) be the function to be maximized or minimized, and the constraint function is:

\[
g(x, y, z) = \frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} - 1 = 0.
\]

The Lagrangian function is defined as:

\[
\mathcal{L}(x, y, z, \lambda) = f(x, y, z) - \lambda g(x, y, z).
\]

This becomes:

\[
\mathcal{L}(x, y, z, \lambda) = (x + y + z) - \lambda \left( \frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} - 1 \right).
\]

### Step 2: Compute partial derivatives
We need to find the critical points by setting the partial derivatives of the Lagrange function with respect to \(x\), \(y\), \(z\), and \(\lambda\) to zero.

- \(\frac{\partial \mathcal{L}}{\partial x} = 1 - \lambda \cdot \frac{x}{2} = 0\)
- \(\frac{\partial \mathcal{L}}{\partial y} = 1 - \lambda \cdot \frac{2y}{5} = 0\)
- \(\frac{\partial \mathcal{L}}{\partial z} = 1 - \lambda \cdot \frac{2z}{25} = 0\)
- \(\frac{\partial \mathcal{L}}{\partial \lambda} = \frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} - 1 = 0\).

### Step 3: Solve the system of equations
We now solve the system of equations obtained from the partial derivatives. Let me work through the calculations.The solutions to the system of equations are:

1. \( x = -\frac{2\sqrt{34}}{17}, y = -\frac{5\sqrt{34}}{34}, z = -\frac{25\sqrt{34}}{34} \)
2. \( x = \frac{2\sqrt{34}}{17}, y = \frac{5\sqrt{34}}{34}, z = \frac{25\sqrt{34}}{34} \)

### Step 4: Calculate \(x + y + z\) for each solution
- For the first solution: 
  \[
  x + y + z = -\frac{2\sqrt{34}}{17} - \frac{5\sqrt{34}}{34} - \frac{25\sqrt{34}}{34}
  \]
- For the second solution: 
  \[
  x + y + z = \frac{2\sqrt{34}}{17} + \frac{5\sqrt{34}}{34} + \frac{25\sqrt{34}}{34}
  \]

Let me calculate these values.The values of \( x + y + z \) are:

- Minimum value: \( -5.83 \)
- Maximum value: \( 5.83 \)

Thus, the maximum value of \( x + y + z \) is approximately \( 5.83 \), and the minimum value is approximately \( -5.83 \), subject to the given constraint.

Would you like further details or clarifications on any part of the solution?

Here are 5 related questions you can explore:
1. What is the geometric interpretation of using Lagrange multipliers?
2. How does the method of Lagrange multipliers work in cases with more constraints?
3. Can the maximum and minimum values of \( x + y + z \) be calculated using different methods?
4. How would the problem change if the constraint were different (e.g., an inequality)?
5. What happens to the optimization problem if one of the variables, say \( z \), is held constant?

**Tip:** When solving optimization problems with constraints, always ensure the constraint is active (equality holds) at the critical points to avoid missing solutions.

Find the maximum and minimum values of x + y + z subject to the constraint \(\frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} = 1\).

Learn how to find the maximum and minimum values of x + y + z under the constraint \(\frac{x^2}{4} + \frac{y^2}{5} + \frac{z^2}{25} = 1\). This example uses the method of Lagrange multipliers to solve the problem, with detailed step-by-step calculations.

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