To find the value of 1/M, we need to first find the maximum value of the function xyz^2, given the constraint x + 2y + 3z = 1.

Constraint:
Given: x + 2y + 3z = 1

Objective:
To find the maximum value of xyz^2.

Method:
To solve this problem, we can use the method of Lagrange multipliers. This method allows us to find the maximum or minimum values of a function subject to constraints. In this case, our function is xyz^2 and our constraint is x + 2y + 3z = 1.

Step 1: Formulating the Lagrangian function:
The Lagrangian function is defined as the objective function minus the product of the constraint equation and a Lagrange multiplier (λ).

L(x, y, z, λ) = xyz^2 - λ(x + 2y + 3z - 1)

Step 2: Finding the partial derivatives:
Next, we need to find the partial derivatives of the Lagrangian function with respect to x, y, z, and λ.

∂L/∂x = yz^2 - λ
∂L/∂y = xz^2 - 2λ
∂L/∂z = 2xyz - 3λ
∂L/∂λ = -(x + 2y + 3z - 1)

Step 3: Setting the partial derivatives equal to zero:
To find the critical points, we set the partial derivatives equal to zero and solve the resulting system of equations.

yz^2 - λ = 0
xz^2 - 2λ = 0
2xyz - 3λ = 0
x + 2y + 3z - 1 = 0

Step 4: Solving the system of equations:
Solving the system of equations will give us the critical points.

From the first equation, we can solve for λ in terms of y and z:
λ = yz^2

Substituting this value of λ into the second equation, we get:
xz^2 - 2yz^2 = 0
xz^2 = 2yz^2
x = 2y

Substituting these values of x and λ into the third equation, we get:
2xy^2 - 3yz^2 = 0
2y^3 - 3yz^2 = 0
2y^2 - 3z^2 = 0

Substituting the value of x = 2y into the fourth equation, we get:
2y + 2y + 3z - 1 = 0
4y + 3z - 1 = 0
4y = 1 - 3z
y = (1 - 3z)/4

Substituting this value of y into the third equation, we get:
2((1 - 3z)/4)^2 - 3z^2 = 0
(1 - 3z)^2 - 6z^2 = 0
1 - 6z + 9z^2 -