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How could you use the implicit function theorem to prove something like this?

  1. Aug 5, 2009 #1
    Suppose you know that a function [itex]g(x,y,z)[/itex] has a unique, non-degenerate minimum at some point [itex](x_0,y_0,z_0)[/itex]. How could you go about using the implicit function theorem to prove that [itex]f(x,y,z) = g(x,y,z) + C h(x,y,z)[/itex], where [itex]C[/itex] is some constant, has a minimum at some point [itex](x_c,y_c,z_c)[/itex]? Could we further show that [itex](x_c,y_c,z_c)[/itex] would need to be close to [itex](x_0,y_0,z_0)[/itex]?

    I realize that I've left so many things unspecified that a lot of details will have to be omitted. But could you walk me through the basic program for proving something like this (e.g., things I'd have to show about the function [itex]h[/itex] or the constant [itex]C[/itex])?
     
  2. jcsd
  3. Aug 5, 2009 #2
    I suppose "non-degenerate" means the Hessian is positive definite? Then you need to diagonalize Hessian matrix and take C related to its eigenvalues.
     
  4. Aug 5, 2009 #3
    Hehe...yeah, that's the thing...I'm not really sure what a "non-degenerate" minimum is. Is that kind of an accepted definition for non-degenerate minimum?
     
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