In this question, we are required to find the minimum value of X*2 Y*2 Z*2 subject to the condition that XYZ=A*3, where A is a constant. This problem can be solved using the method of Lagrange multipliers.


The method of Lagrange multipliers is a powerful tool for solving constrained optimization problems. It involves the introduction of a new variable, called the Lagrange multiplier, to incorporate the constraint into the objective function. The Lagrange multiplier is then used to find the critical points of the function subject to the constraint.


Let f(X,Y,Z) = X*2 Y*2 Z*2 be the objective function and g(X,Y,Z) = XYZ - A*3 be the constraint function. We introduce a Lagrange multiplier λ and form the Lagrangian function L(X,Y,Z,λ) = X*2 Y*2 Z*2 - λ(XYZ - A*3).

Taking the partial derivatives of L with respect to X, Y, Z, and λ, we get:

∂L/∂X = 2XY*2 Z*2 - λYZ
∂L/∂Y = 2X*2 YZ*2 - λXZ
∂L/∂Z = 2X*2 Y*2 Zλ - λXY
∂L/∂λ = XYZ - A*3

Setting these equations equal to zero and solving for X, Y, Z, and λ, we obtain the critical points of the function subject to the constraint.

After solving the equations we get X = Y = Z = (A/3)^(1/3), and λ = 2(A/3)^(-2/3)

Finally, we substitute these values into the objective function to obtain the minimum value of X*2 Y*2 Z*2 subject to the constraint XYZ = A*3. The minimum value is (A/3)^(2/3)*2^(3/2).


In conclusion, we have used the method of Lagrange multipliers to find the minimum value of X*2 Y*2 Z*2 subject to the constraint XYZ = A*3. The minimum value is (A/3)^(2/3)*2^(3/2). This problem illustrates the power of the Lagrangian method in solving constrained optimization problems.

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