Use Lagrange's method to find maximum.

1. Feb 7, 2015

MechanicalBrank

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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.

2. Feb 7, 2015

Ray Vickson

So, do I understand correctly that you want to maximize the function
$$F(x,y) = \left(\frac{\partial f(x,y)}{\partial x}\right)^2 + \left(\frac{\partial f(x,y)}{\partial y}\right)^2 \; ?$$
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. Feb 7, 2015

MechanicalBrank

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.