find the maximum value of the function f(x,y,z) = x+2z under the two constraints of (x^2)+2(y^2)=10 and (x^2)+(y^2)+4(z^2)=8

We are tasked with finding the maximum value of the function \( f(x, y, z) = x + 2z \) under the constraints:

\[
g_1(x, y) = x^2 + 2y^2 = 10
\]
\[
g_2(x, y, z) = x^2 + y^2 + 4z^2 = 8
\]

### Step 1: Lagrange Multipliers
To solve this, we apply the method of Lagrange multipliers. We need to find \( \nabla f \), \( \nabla g_1 \), and \( \nabla g_2 \), where \( \nabla \) denotes the gradient.

1. \( \nabla f(x, y, z) = \left( 1, 0, 2 \right) \)
2. \( \nabla g_1(x, y) = \left( 2x, 4y, 0 \right) \)
3. \( \nabla g_2(x, y, z) = \left( 2x, 2y, 8z \right) \)

We now set up the Lagrange multiplier equations:

\[
\nabla f = \lambda \nabla g_1 + \mu \nabla g_2
\]

This gives the system of equations:

1. \( 1 = \lambda (2x) + \mu (2x) \)
2. \( 0 = \lambda (4y) + \mu (2y) \)
3. \( 2 = \lambda (0) + \mu (8z) \)

### Step 2: Solving the system

From the second equation:

\[
0 = \lambda (4y) + \mu (2y)
\]

For this to hold, either \( y = 0 \) or \( \lambda (4) + \mu (2) = 0 \). Let's consider both cases.

#### Case 1: \( y = 0 \)

If \( y = 0 \), the constraints become:

1. \( x^2 = 10 \), so \( x = \pm \sqrt{10} \)
2. \( x^2 + 4z^2 = 8 \), substituting \( x^2 = 10 \):

\[
10 + 4z^2 = 8 \quad \Rightarrow \quad 4z^2 = -2
\]

This yields no real solution for \( z \). Therefore, \( y = 0 \) does not lead to a valid solution.

#### Case 2: \( \lambda (4) + \mu (2) = 0 \)

This implies:

\[
\lambda = -\frac{\mu}{2}
\]

Substitute this into the other two equations:

1. \( 1 = \lambda (2x) + \mu (2x) \Rightarrow 1 = (-\frac{\mu}{2})(2x) + \mu(2x) = \mu x \)
2. \( 2 = \mu(8z) \Rightarrow \mu = \frac{1}{4z} \)

Substitute \( \mu = \frac{1}{4z} \) into \( 1 = \mu x \):

\[
1 = \frac{x}{4z} \quad \Rightarrow \quad x = 4z
\]

### Step 3: Substituting into the constraints

Now, substitute \( x = 4z \) into the constraints:

1. \( (4z)^2 + 2y^2 = 10 \Rightarrow 16z^2 + 2y^2 = 10 \Rightarrow 2y^2 = 10 - 16z^2 \Rightarrow y^2 = 5 - 8z^2 \)
2. \( (4z)^2 + y^2 + 4z^2 = 8 \Rightarrow 16z^2 + y^2 + 4z^2 = 8 \Rightarrow 20z^2 + y^2 = 8 \)

Substitute \( y^2 = 5 - 8z^2 \) into this equation:

\[
20z^2 + (5 - 8z^2) = 8 \quad \Rightarrow \quad 12z^2 + 5 = 8 \quad \Rightarrow \quad 12z^2 = 3 \quad \Rightarrow \quad z^2 = \frac{1}{4}
\]

So, \( z = \pm \frac{1}{2} \).

### Step 4: Find corresponding values for \( x \) and \( y \)

If \( z = \frac{1}{2} \), then \( x = 4z = 2 \).

Substitute \( z = \frac{1}{2} \) into \( y^2 = 5 - 8z^2 \):

\[
y^2 = 5 - 8\left(\frac{1}{2}\right)^2 = 5 - 2 = 3 \quad \Rightarrow \quad y = \pm \sqrt{3}
\]

### Step 5: Maximum value of \( f(x, y, z) \)

The function is \( f(x, y, z) = x + 2z \). Substituting \( x = 2 \) and \( z = \frac{1}{2} \):

\[
f(x, y, z) = 2 + 2\left(\frac{1}{2}\right) = 3
\]

Thus, the maximum value of \( f(x, y, z) = 3 \).

Would you like further details or clarifications? Here are some related questions to explore:

1. How does the method of Lagrange multipliers work in multivariable optimization?
2. What are the geometric interpretations of constraints in optimization problems?
3. Can the method of Lagrange multipliers solve problems with inequality constraints?
4. How does the solution change if the constraints are relaxed or modified?
5. What are the conditions for a local maximum or minimum in constrained optimization?

**Tip:** Always check for boundary conditions or special cases, like zero values, when solving optimization problems.

Find the maximum value of the function f(x,y,z) = x + 2z under the two constraints of (x^2) + 2(y^2) = 10 and (x^2) + (y^2) + 4(z^2) = 8.

Learn how to maximize the function f(x, y, z) = x + 2z under the constraints x² + 2y² = 10 and x² + y² + 4z² = 8 using the method of Lagrange multipliers. This problem involves key concepts from multivariable calculus and optimization, and it is suitable for advanced mathematics students.

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