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We need to find the minimum value of the function:

\[
f(x, y, z) = x^2 + y^2 + z^2
\]

subject to the constraints:

\[
g(x, y, z) = x + y + z - 8 = 0
\]
\[
h(x, y, z) = 2x - y + 3z - 28 = 0
\]

To solve this, we can use the method of **Lagrange multipliers**. This involves introducing two Lagrange multipliers, \(\lambda\) and \(\mu\), for each constraint. Then, we set up the following system of equations:

1. The gradients must satisfy:
   \[
   \nabla f = \lambda \nabla g + \mu \nabla h
   \]
   
2. The constraints must hold:
   \[
   g(x, y, z) = 0
   \]
   \[
   h(x, y, z) = 0
   \]

### Step 1: Calculate Gradients

The gradient of \(f\) is:
\[
\nabla f = (2x, 2y, 2z)
\]

The gradient of \(g\) is:
\[
\nabla g = (1, 1, 1)
\]

The gradient of \(h\) is:
\[
\nabla h = (2, -1, 3)
\]

### Step 2: Set Up the System of Equations

Using the Lagrange multiplier equation \(\nabla f = \lambda \nabla g + \mu \nabla h\), we get:

1. For \(x\)-component:
   \[
   2x = \lambda \cdot 1 + \mu \cdot 2
   \]
   which simplifies to:
   \[
   2x = \lambda + 2\mu
   \]

2. For \(y\)-component:
   \[
   2y = \lambda \cdot 1 + \mu \cdot (-1)
   \]
   which simplifies to:
   \[
   2y = \lambda - \mu
   \]

3. For \(z\)-component:
   \[
   2z = \lambda \cdot 1 + \mu \cdot 3
   \]
   which simplifies to:
   \[
   2z = \lambda + 3\mu
   \]

### Step 3: Solve the System with Constraints

We also have the constraint equations:

\[
x + y + z = 8
\]
\[
2x - y + 3z = 28
\]

Now, we solve this system of equations. Let me work through the algebra.The solution to the system is:

\[
x = 4, \quad y = -2, \quad z = 6
\]

Now, we can substitute these values into the function \( f(x, y, z) = x^2 + y^2 + z^2 \) to find the minimum value.

### Step 4: Calculate \( f(x, y, z) \) at the Solution Point

Let's substitute \( x = 4 \), \( y = -2 \), and \( z = 6 \) into \( f(x, y, z) \).The minimum value of the function \( f(x, y, z) = x^2 + y^2 + z^2 \) subject to the given constraints is:

\[
f(4, -2, 6) = 56
\]

Would you like further details on any steps or have any additional questions?

### Related Questions

1. How does the method of Lagrange multipliers work in general?
2. What other methods can be used to solve constrained optimization problems?
3. How would the solution change if we modified the function \( f(x, y, z) \)?
4. Can we solve this problem using matrix methods?
5. What would happen if one of the constraints was non-linear?

### Tip

When using Lagrange multipliers, always check that the solution satisfies both the constraints and the original function's minimum requirements.

Tentukan nilai minimum dari fungsi: f(x, y, z) = x^2 + y^2 + z^2 dengan kendala: g(x, y, z) = x + y + z - 8 dan h(x, y, z) = 2x - y + 3z - 28

This page provides a step-by-step solution for finding the minimum value of the function f(x, y, z) = x^2 + y^2 + z^2, subject to constraints g(x, y, z) = x + y + z - 8 and h(x, y, z) = 2x - y + 3z - 28. It covers the method of Lagrange multipliers and is suitable for undergraduate-level multivariable calculus students.

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