you have note sate the corrcet sign for constraints in Lagrange function, correct that please

Thank you for pointing that out. You're right; the correct signs in the Lagrange function are crucial for solving the problem accurately. Let's correct the Lagrange function and go through the process again.

### Problem Summary:
- **Objective:** Maximize \( f(x, y, z) = x + 2z \)
- **Subject to the constraints:**
  \[
  g_1(x, y, z) = x + y + z - 1 = 0
  \]
  \[
  g_2(x, y, z) = x^2 + y^2 + z^2 - \frac{7}{4} = 0
  \]

### Step 1: Present the Correct Lagrange Function
The correct Lagrange function \( \mathcal{L} \) with the multipliers \( \lambda \) and \( \mu \) should be:
\[
\mathcal{L}(x, y, z, \lambda, \mu) = x + 2z + \lambda (1 - x - y - z) + \mu \left(\frac{7}{4} - x^2 - y^2 - z^2\right)
\]

### Step 2: Find the First Order Conditions
Now, let's find the first-order conditions by taking the partial derivatives of \( \mathcal{L} \) with respect to each variable and setting them equal to zero:

1. **Partial derivative with respect to \( x \):**
   \[
   \frac{\partial \mathcal{L}}{\partial x} = 1 - \lambda - 2\mu x = 0
   \]

2. **Partial derivative with respect to \( y \):**
   \[
   \frac{\partial \mathcal{L}}{\partial y} = -\lambda - 2\mu y = 0
   \]

3. **Partial derivative with respect to \( z \):**
   \[
   \frac{\partial \mathcal{L}}{\partial z} = 2 - \lambda - 2\mu z = 0
   \]

4. **Partial derivative with respect to \( \lambda \):**
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = 1 - x - y - z = 0
   \]

5. **Partial derivative with respect to \( \mu \):**
   \[
   \frac{\partial \mathcal{L}}{\partial \mu} = \frac{7}{4} - x^2 - y^2 - z^2 = 0
   \]

### Step 3: Solve for Optimal Values \( x^*, y^*, z^* \)
Let's rewrite the system of equations derived from the first-order conditions:

1. \( 1 = \lambda + 2\mu x \) \quad (10)
2. \( \lambda = -2\mu y \) \quad (11)
3. \( 2 = \lambda + 2\mu z \) \quad (12)
4. \( x + y + z = 1 \) \quad (13)
5. \( x^2 + y^2 + z^2 = \frac{7}{4} \) \quad (14)

### Solving the Equations:
- From (11): \( \lambda = -2\mu y \)
- Substitute into (10) and (12):
  \[
  1 = -2\mu y + 2\mu x \quad \text{or} \quad \mu(x - y) = \frac{1}{2} \quad \Rightarrow \quad x - y = \frac{1}{2\mu} \quad \text{(15)}
  \]
  \[
  2 = -2\mu y + 2\mu z \quad \text{or} \quad \mu(z - y) = 1 \quad \Rightarrow \quad z - y = \frac{1}{\mu} \quad \text{(16)}
  \]
- From (15) and (16):
  \[
  x = y + \frac{1}{2\mu}, \quad z = y + \frac{1}{\mu}
  \]
- Substitute these into the constraint \( x + y + z = 1 \) from (13):
  \[
  y + \frac{1}{2\mu} + y + y + \frac{1}{\mu} = 1 \quad \Rightarrow \quad 3y + \frac{3}{2\mu} = 1
  \quad \Rightarrow \quad y = \frac{1}{3} - \frac{1}{2\mu}
  \]
- Using this \( y \)

Explore the importance of correct signs in the Lagrange function through a detailed problem-solving approach. Learn how to maximize a function subject to given constraints using Lagrange multipliers. Suitable for advanced undergraduate students and anyone interested in optimization problems.

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