To maximize $u = \frac{5xyz}{x + 2y + 4z}$ given the constraint $xyz = 8$, we use techniques from optimization, such as Lagrange multipliers.

Step 1: Problem Setup

Define: $f(x, y, z) = \frac{5xyz}{x + 2y + 4z}, \quad g(x, y, z) = xyz - 8.$ The goal is to maximize $f(x, y, z)$ subject to $g(x, y, z) = 0$.

Step 2: Lagrange Multiplier Conditions

Introduce a Lagrange multiplier $\lambda$ and solve: $\nabla f = \lambda \nabla g.$

• The gradient of $f(x, y, z)$ is: $\nabla f = \left( \frac{5yz(x + 2y + 4z) - 5xyz}{(x + 2y + 4z)^2}, \frac{5xz(x + 2y + 4z) - 10xyz}{(x + 2y + 4z)^2}, \frac{5xy(x + 2y + 4z) - 20xyz}{(x + 2y + 4z)^2} \right).$
• The gradient of $g(x, y, z)$ is: $\nabla g = (yz, xz, xy).$

Equating $\nabla f = \lambda \nabla g$, we obtain: $\frac{5yz(x + 2y + 4z) - 5xyz}{(x + 2y + 4z)^2} = \lambda yz,$ $\frac{5xz(x + 2y + 4z) - 10xyz}{(x + 2y + 4z)^2} = \lambda xz,$ $\frac{5xy(x + 2y + 4z) - 20xyz}{(x + 2y + 4z)^2} = \lambda xy.$

Step 3: Simplify and Solve

From symmetry, assume $x = y = z$ for simplicity. Then $xyz = 8$ implies: $x^3 = 8 \implies x = 2.$ Thus, $x = y = z = 2$.

Step 4: Verify the Solution

Substitute $x = y = z = 2$ into $u$: $u = \frac{5(2)(2)(2)}{2 + 2(2) + 4(2)} = \frac{5 \cdot 8}{2 + 4 + 8} = \frac{40}{14} = \frac{20}{7}.$

Final Answer:

The value of $u$ is maximized at $x = 2$, $y = 2$, $z = 2$, and $u = \frac{20}{7}$.

Would you like detailed derivations of specific steps or further clarifications?

Follow-Up Questions:

1. Can we maximize $u$ for other values if $x, y, z$ are not symmetric?
2. How does the choice of method (Lagrange multipliers) affect the solution?
3. What happens if the constraint $xyz = 8$ changes?
4. Are there numerical methods to verify the solution computationally?
5. Why does symmetry simplify the optimization problem here?

Tip:

Symmetry in constraints often simplifies optimization problems significantly, so test symmetric solutions first when applicable!