Use Lagrange's method to find maximum.

  • #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.
 

Answers and Replies

  • #2
Ray Vickson
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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
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.
 

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