Proving extrema using taylor series and Hessian Matrix

In summary: From negative definite we can conclude f_xx and f_yy are negative but the same cannot be concluded for f_xy. Thus how do we conclude that when the Hessian is negative the expansion becomes f(P) minus something. Is it possible for the terms behind f(P) to add up to a positive number if f_xy is large (and positive) in the expansion?
  • #1
dispiriton
8
0
How do I use Taylor Series to show f(P) is a local maximum at a stationary point P if the Hessian matrix is negative definite.

I understand that some of the coefficients of the terms of the taylor series expansion are the coordinates of the Hessian matrix but for the f_xy term there is no conclusion and how does it affect the Taylor expansion to show it is a maximum?
 
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  • #2
Write the second order Taylor polynomial at P. It should be of the form
[tex]f(P)+ f_xx(P)(x- x_0)^2+ 2f_{xy}(x- x_0)(y- y_0)+ f_{yy}(y- y_0)^2[/tex]

And if the Hessian is negative definite, you can write that as [itex]f(P)[/itex] minus a square.
 
  • #3
HallsofIvy said:
Write the second order Taylor polynomial at P. It should be of the form
[tex]f(P)+ f_xx(P)(x- x_0)^2+ 2f_{xy}(x- x_0)(y- y_0)+ f_{yy}(y- y_0)^2[/tex]

And if the Hessian is negative definite, you can write that as [itex]f(P)[/itex] minus a square.

But when the Hessian is negative definite we can only say that f_xx and f_yy are negative. Can the same be concluded for the f_xy term?
 
  • #4
dispiriton said:
But when the Hessian is negative definite we can only say that f_xx and f_yy are negative.k
No, that is not true- I suggest you review the definitions of "positive definite" and "negative definite".

Can the same be concluded for the f_xy term?
 
  • #5
HallsofIvy said:
No, that is not true- I suggest you review the definitions of "positive definite" and "negative definite".

From negative definite we can conclude f_xx and f_yy are negative but the same cannot be concluded for f_xy. Thus how do we conclude that when the Hessian is negative the expansion becomes f(P) minus something. Is it possible for the terms behind f(P) to add up to a positive number if f_xy is large (and positive)
 

1. What is the purpose of using a Taylor series and Hessian Matrix to prove extrema?

The Taylor series and Hessian Matrix are mathematical tools used to analyze the behavior of a function at a particular point. They can help determine if a function has a maximum or minimum value at that point, which is useful in optimization problems and in understanding the overall behavior of a function.

2. How do you use a Taylor series to prove the existence of extrema?

To use a Taylor series to prove the existence of extrema, you must first find the critical points of the function by setting its derivative equal to zero. Then, you can plug these critical points into the Taylor series and analyze the behavior of the function at those points. If the function has a maximum or minimum value at a critical point, this can be confirmed using the Taylor series.

3. What is the Hessian Matrix and how does it relate to proving extrema?

The Hessian Matrix is a square matrix of second-order partial derivatives of a function. It can be used to determine the nature of the critical points of a function, which can help in proving the existence of extrema. If the Hessian Matrix is positive definite, the function has a minimum value at the critical point. If it is negative definite, the function has a maximum value. If it is indefinite, the critical point is a saddle point.

4. How does the accuracy of the Taylor series and Hessian Matrix affect the proof of extrema?

The accuracy of the Taylor series and Hessian Matrix is crucial in determining the existence and nature of extrema. If the Taylor series is not accurate enough, it may not accurately represent the behavior of the function at the critical point. Similarly, if the Hessian Matrix is not calculated correctly, it may lead to incorrect conclusions about the nature of the critical point.

5. Are there any limitations to using the Taylor series and Hessian Matrix to prove extrema?

Yes, there are some limitations to using the Taylor series and Hessian Matrix. They may not be applicable to all functions, especially those that are not differentiable or have discontinuities. Additionally, finding the critical points and calculating the Hessian Matrix can be a time-consuming process for more complex functions.

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