Problem: Suppose x, y, z are positive real numbers such that x + 2y + 3z = 1. If M is the maximum value of xyz^2, then the value of 1/M is?

Solution:

Introduction:
We are given a constraint x + 2y + 3z = 1 and we need to find the maximum value of M = xyz^2. In order to find this maximum value, we will use the method of Lagrange Multipliers.

Step 1: Define the Objective Function:
Let f(x, y, z) = xyz^2 be the objective function.

Step 2: Define the Constraint Function:
Let g(x, y, z) = x + 2y + 3z - 1 be the constraint function.

Step 3: Form the Lagrangian Function:
The Lagrangian function L(x, y, z, λ) is defined as:
L(x, y, z, λ) = f(x, y, z) - λg(x, y, z)

In our case, the Lagrangian function is:
L(x, y, z, λ) = xyz^2 - λ(x + 2y + 3z - 1)

Step 4: Find the Partial Derivatives:
We need to find the partial derivatives of L(x, y, z, λ) 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 5: Set the Partial Derivatives 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 ...(1)
xz^2 - 2λ = 0 ...(2)
2xyz - 3λ = 0 ...(3)
x + 2y + 3z - 1 = 0 ...(4)

Step 6: Solve the System of Equations:
From equation (1), we have:
yz^2 = λ ...(5)

From equation (2), we have:
xz^2 = 2λ ...(6)

Dividing equation (6) by equation (5), we get:
x/y = 2z ...(7)

From equation (3), we have:
2xyz = 3λ ...(8)

Substituting equation (8) into equation (6), we get:
3xz^2 = 2λ ...(9)

Dividing equation (9) by equation (5), we get:
3x/y = 2z ...(10)

Equating equations (7) and (10), we get:
x/y = 3x/2y
2y = 3x
x = 2y/3 ...(11)

Substituting equation (11) into equation (4), we get:
2y/3 + 2