Introduction
In this problem, we are required to divide a given number n into three parts x, y, and z such that the value of ayz bzx cxy is maximum or minimum. Here, a, b, and c are constants.

Solution
To solve this problem, we need to follow the following steps:

Step 1: Write the Objective Function
The objective function is given as:
F(x, y, z) = ayz - bzx + cxy

Step 2: Write the Constraints
The constraints are given as:
x + y + z = n
x, y, z ≥ 0

Step 3: Solve the Problem
To solve this problem, we need to use the method of Lagrange Multipliers.

We first write the Lagrangian function as:
L(x, y, z, λ) = ayz - bzx + cxy + λ(x + y + z - n)

Next, we find the partial derivatives of L with respect to x, y, z, and λ and set them equal to zero.

∂L/∂x = cy - bz + λ = 0
∂L/∂y = cx + λ = 0
∂L/∂z = ay - bx + λ = 0
∂L/∂λ = x + y + z - n = 0

Solving these equations, we get:
x = (b - c)λ/(2a)
y = -λ/c
z = (b + c)λ/(2a)

Substituting the values of x, y, and z in the constraint equation x + y + z = n, we get:
λ = 2an/(b + c - 2c)

Therefore, the values of x, y, and z are given as:
x = n(b - c)/(b + c - 2c)
y = -2an/c(b + c - 2c)
z = n(b + c)/(b + c - 2c)

Step 4: Interpret the Results
The value of F(x, y, z) can now be calculated using the values of x, y, and z. If ayz bzx cxy > 0, then the function is maximized, and if ayz bzx cxy < 0,="" then="" the="" function="" is="" />

Conclusion
In this problem, we used the method of Lagrange Multipliers to find the values of x, y, and z that maximize or minimize the function ayz bzx cxy. The solution involves finding the partial derivatives of the Lagrangian function and solving the resulting equations.

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