Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.)
f(x, y, z) = xyz;    x2 + 2y2 + 3z2 = 96
maximum	    	
  
Incorrect: Your answer is incorrect.
minimum	

We are tasked with finding the maximum and minimum values of the function \( f(x, y, z) = xyz \) subject to the constraint \( x^2 + 2y^2 + 3z^2 = 96 \) using the method of Lagrange multipliers.

### Step 1: Define the Lagrange function
We introduce the Lagrange multiplier \( \lambda \) and form the Lagrange function:
\[
\mathcal{L}(x, y, z, \lambda) = xyz + \lambda (96 - x^2 - 2y^2 - 3z^2)
\]
where the constraint is \( g(x, y, z) = x^2 + 2y^2 + 3z^2 - 96 = 0 \).

### Step 2: Compute the gradients
We need to find the partial derivatives of \( \mathcal{L} \) with respect to \( x \), \( y \), \( z \), and \( \lambda \).

1. \( \frac{\partial \mathcal{L}}{\partial x} = yz - 2\lambda x = 0 \)
2. \( \frac{\partial \mathcal{L}}{\partial y} = xz - 4\lambda y = 0 \)
3. \( \frac{\partial \mathcal{L}}{\partial z} = xy - 6\lambda z = 0 \)
4. \( \frac{\partial \mathcal{L}}{\partial \lambda} = 96 - x^2 - 2y^2 - 3z^2 = 0 \)

### Step 3: Solve the system of equations
We now solve the system of equations:
\[
yz = 2\lambda x
\]
\[
xz = 4\lambda y
\]
\[
xy = 6\lambda z
\]
\[
x^2 + 2y^2 + 3z^2 = 96
\]

To approach this, we can look for relationships between \( x \), \( y \), and \( z \). Start by dividing equations to eliminate \( \lambda \).

#### 1. Divide equation 1 by equation 2:
\[
\frac{yz}{xz} = \frac{2\lambda x}{4\lambda y} \implies \frac{y}{x} = \frac{1}{2} \implies y = \frac{x}{2}
\]

#### 2. Divide equation 2 by equation 3:
\[
\frac{xz}{xy} = \frac{4\lambda y}{6\lambda z} \implies \frac{z}{y} = \frac{2}{3} \implies z = \frac{2y}{3}
\]

#### 3. Substitute \( y = \frac{x}{2} \) and \( z = \frac{2y}{3} \) into the constraint
Substitute these into the constraint equation \( x^2 + 2y^2 + 3z^2 = 96 \).

Since \( z = \frac{2y}{3} \), first substitute \( z \) into the equation:
\[
x^2 + 2y^2 + 3\left(\frac{2y}{3}\right)^2 = 96
\]
\[
x^2 + 2y^2 + 3\left(\frac{4y^2}{9}\right) = 96
\]
\[
x^2 + 2y^2 + \frac{4y^2}{3} = 96
\]
Multiply through by 3 to eliminate the fraction:
\[
3x^2 + 6y^2 + 4y^2 = 288
\]
\[
3x^2 + 10y^2 = 288
\]
Since \( y = \frac{x}{2} \), substitute \( y = \frac{x}{2} \) into the equation:
\[
3x^2 + 10\left(\frac{x}{2}\right)^2 = 288
\]
\[
3x^2 + 10\left(\frac{x^2}{4}\right) = 288
\]
\[
3x^2 + \frac{10x^2}{4} = 288
\]
Multiply through by 4 to clear the fraction:
\[
12x^2 + 10x^2 = 1152
\]
\[
22x^2 = 1152
\]
\[
x^2 = \frac{1152}{22} = 52.36
\]
Thus,
\[
x = \sqrt{52.36} \approx 7.23
\]

### Step 4: Calculate \( y \) and \( z \)
Now, \( y = \frac{x}{2} \approx \frac{7.23}{2} \approx 3.62 \), and \( z = \frac{2y}{3} \approx \frac{2(3.62)}{3} \approx 2.41 \).

### Step 5: Calculate the function value \( f(x, y, z) = xyz \)
\[
f(7.23, 3.62, 2.41) = 7.23 \times 3.62 \times 2.41 \approx 63.02
\]

Thus, the **maximum value** of \( f(x, y, z) \) is approximately 63.02.

### Step 6: Check for minimum
By the nature of the function \( f(x, y, z) = xyz \), which has no lower bound (since it can take negative values as well), the minimum value is **DNE** (Does Not Exist).

### Final Answer:
- Maximum value: 63.02
- Minimum value: DNE

Would you like more details or further explanations?

Here are 5 related questions to deepen your understanding:
1. What are Lagrange multipliers, and how are they used in optimization?
2. How would the problem change if the constraint were different?
3. What are the necessary conditions for a function to have both maximum and minimum values?
4. How can you visualize the constraint as a geometric shape?
5. What is the geometric interpretation of the solution found?

**Tip:** Always check the boundary conditions when using Lagrange multipliers to ensure that all possible extrema are found.

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.) f(x, y, z) = xyz;    x^2 + 2y^2 + 3z^2 = 96

Learn how to apply Lagrange multipliers to find the maximum and minimum values of the function f(x, y, z) = xyz subject to the constraint x^2 + 2y^2 + 3z^2 = 96. This step-by-step solution covers multivariable calculus and optimization techniques, ideal for undergraduate students.

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