Use Lagrange's method to find maximum.

In summary, the problem is to find the general equations for finding the maximum of a function in the form of u^2 + v^2 using Lagrange's method. The solution involves creating a system of equations using the constraints u - ∂f/∂x = 0 and v - ∂f/∂y = 0 and then using the Lagrange multiplier method to optimize the function. This method can then be applied to find the maximum of specific given functions.
  • #1

Homework Statement


Let u = ∂f/∂x and v = ∂f/∂y. Use Lagrange's method to set up the equations necessary for finding the maxium of u^2 + v^2.

Homework Equations

The Attempt at a Solution


I know how to minimize and maximize a function that's a subject to another one. I'm not quite sure what the subject is in this case, pointers would be helpful.
 
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  • #2
MechanicalBrank said:

Homework Statement


Let u = ∂f/∂x and v = ∂f/∂y. Use Lagrange's method to set up the equations necessary for finding the maxium of u^2 + v^2.

Homework Equations

The Attempt at a Solution


I know how to minimize and maximize a function that's a subject to another one. I'm not quite sure what the subject is in this case, pointers would be helpful.

So, do I understand correctly that you want to maximize the function
[tex] F(x,y) = \left(\frac{\partial f(x,y)}{\partial x}\right)^2 + \left(\frac{\partial f(x,y)}{\partial y}\right)^2 \; ? [/tex]
If so, are you given any more information about the function ##f##, because as it stands, the problem is not particularly well-posed: it is possible to give functions ##f## for which your ##F(x,y)## has no maximum---not even any local maxima.

However, assuming the problem is well-posed, you can proceed in two ways: (i) directly, from the formula for ##F##; and (ii) using the conditions ##g_1(x,y,u,v) \equiv u - \partial f /\partial x = 0## and ##g_2(x,y,u,v) \equiv v - \partial f/ \partial y = 0## as constraints in the optimization of ##h(x,y,u,v) = u^2 + v^2## (with ##h## happening to not actually depend on ##x,y## at all!). That is where you could use the Lagrange multiplier method.
 
  • #3
Ray Vickson said:
So, do I understand correctly that you want to maximize the function
[tex] F(x,y) = \left(\frac{\partial f(x,y)}{\partial x}\right)^2 + \left(\frac{\partial f(x,y)}{\partial y}\right)^2 \; ? [/tex]
If so, are you given any more information about the function ##f##, because as it stands, the problem is not particularly well-posed: it is possible to give functions ##f## for which your ##F(x,y)## has no maximum---not even any local maxima.

However, assuming the problem is well-posed, you can proceed in two ways: (i) directly, from the formula for ##F##; and (ii) using the conditions ##g_1(x,y,u,v) \equiv u - \partial f /\partial x = 0## and ##g_2(x,y,u,v) \equiv v - \partial f/ \partial y = 0## as constraints in the optimization of ##h(x,y,u,v) = u^2 + v^2## (with ##h## happening to not actually depend on ##x,y## at all!). That is where you could use the Lagrange multiplier method.

You have understood correctly. The problem asks to find the general equations for finding any potential maxima. I guess it means that they want me to create a general system of equations that would find the maxima if we were given a function. There's a subproblem after this one where they ask to find the maxima for a few given functions (sinx+siny and sinx*siny) using the method from the first problem.
 

1. What is Lagrange's method?

Lagrange's method, also known as the method of Lagrange multipliers, is a mathematical technique used to find the maximum or minimum value of a function subject to certain constraints. It involves setting up a system of equations using the method of partial derivatives.

2. How does Lagrange's method work?

Lagrange's method involves setting up a system of equations using the method of partial derivatives. The equations are solved simultaneously to find the values of the variables that satisfy both the original function and the given constraints. The maximum or minimum value of the function can then be found using these values.

3. When should Lagrange's method be used?

Lagrange's method should be used when trying to find the maximum or minimum value of a function subject to certain constraints. It is particularly useful when the function and constraints are difficult to work with directly, or when there are multiple constraints that need to be taken into account.

4. What are the advantages of using Lagrange's method?

One of the main advantages of using Lagrange's method is that it allows for the optimization of a function subject to multiple constraints. It also provides a systematic approach to solving these types of problems, making it easier to find the maximum or minimum value of a function.

5. Are there any limitations to using Lagrange's method?

While Lagrange's method is a useful technique, it does have some limitations. It may not always provide the global maximum or minimum value of a function, and it can be computationally intensive for more complex problems. Additionally, it may not work for functions with discontinuities or when the constraints are not differentiable.

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