Lagrange multiplier problem

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  • #1
Elliotc
1
0
1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs



2. Function: V(x,y,z) = 2x2y2z = 8xyz
Constraint: x^2+Y^2+z^2=9




3. So far I have gone
Φ= 8xyz + λ(x^2+y^2+z^2 - 9)

∂Φ/∂x = 8yz + 2λx = 0 equation 1
∂Φ/∂y = 8xz + 2λy = 0 equation 2
∂Φ/∂z = 8xy + 2λz = 0 equation 3
x^2 + y^2 + z^2 = 9

so then I have multiplied equation 1 by x, 2 by y, and 3 by z

so I am left with
8xyz + 2λx^2 = 0
8xyz + 2λy^2 = 0
8xyz + 2λz^2 = 0

add these together and
24xyz +2λ(x^2 + y^2 + z^2)
I know that x^2 + y^2 + z^2 =9 and v=8xyz so
3V=-18λ
V=-6λ

Now I have no idea how to go about the next stage, I am struggling to rearrange the equations to get an answer.
 
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  • #2
Elliotc said:
1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs


2. Function: V(x,y,z) = 2x2y2z = 8xyz
Constraint: x^2+Y^2+z^2=9




3. So far I have gone
Φ= 8xyz + λ(x^2+y^2+z^2 - 9)

∂Φ/∂x = 8yz + 2λx = 0 equation 1
∂Φ/∂y = 8xz + 2λy = 0 equation 2
∂Φ/∂z = 8xy + 2λz = 0 equation 3
x^2 + y^2 + z^2 = 9

so then I have multiplied equation 1 by x, 2 by y, and 3 by z

so I am left with
8xyz + 2λx^2 = 0
8xyz + 2λy^2 = 0
8xyz + 2λz^2 = 0

add these together and
24xyz +2λ(x^2 + y^2 + z^2)
I know that x^2 + y^2 + z^2 =9 and v=8xyz so
3V=-18λ
V=-6λ

Now I have no idea how to go about the next stage, I am struggling to rearrange the equations to get an answer.

Your equations 8xyz + 2*lambda*x^2 = 0, etc., imply that x^2, y^2 and z^2 are equal, so it is easy to get them. Of course, x, y and z are thus determined only up to a +- sign, so you still need to think a bit about which choices make sense.

RGV
 

What is the Lagrange multiplier problem?

The Lagrange multiplier problem is a mathematical optimization problem that involves finding the maximum or minimum of a multivariate function subject to one or more constraints. It is named after Joseph-Louis Lagrange, who first described the method in the late 18th century.

When is the Lagrange multiplier method used?

The Lagrange multiplier method is used when solving constrained optimization problems, where the objective function must be maximized or minimized while satisfying one or more constraints. It is commonly used in economics, engineering, and physics.

How does the Lagrange multiplier method work?

The Lagrange multiplier method involves constructing a new function, called the Lagrangian, by adding the product of the constraint(s) and a new variable, known as the Lagrange multiplier, to the original objective function. The optimal solution is then found by setting the partial derivatives of the Lagrangian with respect to all variables equal to zero.

What are the assumptions of the Lagrange multiplier method?

The Lagrange multiplier method assumes that the objective function and constraints are continuous and differentiable, and that the constraints are independent. It also assumes that the optimal solution exists and is unique.

What are some real-world applications of the Lagrange multiplier method?

The Lagrange multiplier method has numerous applications in various fields, including economics (e.g. finding the optimal production level for a firm), engineering (e.g. optimizing the design of a structure), and physics (e.g. determining the minimum energy state of a system). It is also used in machine learning and data analysis to find the best fit for a model given certain constraints.

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