This problem asks to divide 24 into three parts $x$, $y$, and $z$ such that the continued product $x \cdot y^2 \cdot z^3$ is maximized.

Let’s solve this step by step:

Step 1: Define the constraints.

1. $x + y + z = 24$ (since the sum of the three parts is 24).
2. The function to maximize is $P = x \cdot y^2 \cdot z^3$.

Step 2: Solve using the method of Lagrange multipliers.

Introduce a Lagrange multiplier $\lambda$ for the constraint $x + y + z = 24$. Define: $L(x, y, z, \lambda) = x \cdot y^2 \cdot z^3 - \lambda (x + y + z - 24).$

Step 3: Take partial derivatives.

Compute the partial derivatives of $L$ with respect to $x$, $y$, $z$, and $\lambda$, and set them equal to zero:

1. $\frac{\partial L}{\partial x} = y^2 z^3 - \lambda = 0$.
2. $\frac{\partial L}{\partial y} = 2x y z^3 - \lambda = 0$.
3. $\frac{\partial L}{\partial z} = 3x y^2 z^2 - \lambda = 0$.
4. $\frac{\partial L}{\partial \lambda} = -(x + y + z - 24) = 0$.

Step 4: Simplify.

From the first three equations: $\lambda = y^2 z^3, \quad \lambda = 2x y z^3, \quad \lambda = 3x y^2 z^2.$ Equate $y^2 z^3 = 2x y z^3$, divide through by $y z^3$ (assuming $y, z \neq 0$): $y = 2x.$ Equate $y^2 z^3 = 3x y^2 z^2$, divide through by $y^2 z^2$ (assuming $y, z \neq 0$): $z = 3x.$

Step 5: Substitute into the constraint.

Using $y = 2x$ and $z = 3x$ in $x + y + z = 24$: $x + 2x + 3x = 24.$ $6x = 24 \implies x = 4.$ Then: $y = 2x = 8, \quad z = 3x = 12.$

Step 6: Verify the solution.

The three parts are $x = 4$, $y = 8$, $z = 12$. The product $P = x \cdot y^2 \cdot z^3$ becomes: $P = 4 \cdot 8^2 \cdot 12^3 = 4 \cdot 64 \cdot 1728 = 442,368.$

Thus, the maximum product occurs when $x = 4$, $y = 8$, and $z = 12$.

Would you like me to provide a detailed explanation of any step?

Related Questions:

1. How does the method of Lagrange multipliers work in optimization problems?
2. Can you apply this method to divide a different sum (e.g., 36) into three parts for maximum product?
3. What happens if $x, y, z$ are constrained to integers?
4. Can this be solved using numerical methods instead of calculus?
5. How do higher powers (e.g., $x^2, y^3, z^4$) affect the solution?

Tip:

When solving maximization problems with constraints, always check the boundary conditions (e.g., if one of $x, y, z$ equals zero).