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): \quad \Rightarrow \quad y = \frac{1}{3} - \frac{1}{2\mu}$$
• Using this $y$