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

In summary, the conversation discusses using the implicit function theorem to prove that a function f(x,y,z) has a minimum at a point (x_c,y_c,z_c) by adding a constant C multiplied by another function h(x,y,z). The details of the proof, such as the function h and the constant C, are left unspecified and may involve diagonalizing the Hessian matrix and considering positive definite eigenvalues. The definition of a "non-degenerate" minimum is also mentioned as a possible requirement for this proof.
  • #1
AxiomOfChoice
533
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])?
 
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  • #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.
 
  • #3
g_edgar said:
I suppose "non-degenerate" means the Hessian is positive definite? Then you need to diagonalize Hessian matrix and take C related to its eigenvalues.
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?
 

1. How does the implicit function theorem work?

The implicit function theorem is a powerful tool that allows us to find solutions to equations that cannot be easily solved explicitly. It states that if we have a system of equations, we can use the theorem to find a relation between the variables that allows us to solve for one variable in terms of the others.

2. Can the implicit function theorem be used to prove the existence of solutions?

Yes, the implicit function theorem can be used to prove the existence of solutions to a system of equations. It guarantees that if certain conditions are met, there will be a unique solution to the equations.

3. What are the conditions that must be met in order to use the implicit function theorem?

The main condition for using the implicit function theorem is that the equations must be sufficiently differentiable. This means that they must have continuous partial derivatives, and that the Jacobian matrix must be invertible at the point of interest.

4. How can the implicit function theorem be applied in real-world situations?

The implicit function theorem has many applications in various fields of science and engineering. It can be used to solve optimization problems, find solutions to differential equations, and analyze complex systems in economics and physics.

5. Is the implicit function theorem always applicable?

No, the implicit function theorem is not always applicable. It requires certain conditions to be met, and there are cases where these conditions are not satisfied. In these situations, alternative methods must be used to solve the equations.

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