How to Find a Region Where V'(x,y) < 0?

• Somefantastik
In summary, a negative definite 'hood is a mathematical concept that refers to a region in a vector space where all points have a negative curvature in all directions. It is the opposite of a positive definite 'hood, which has a positive curvature in all directions. Finding a negative definite 'hood is significant in optimization problems, as it helps identify the maximum value of a function. This can be useful in various applications such as machine learning. A negative definite 'hood can be identified by determining the eigenvalues of the Hessian matrix of a function or by using the second derivative test. It is possible for a function to have both a negative and positive definite 'hood at different points, but not at the same point.
Somefantastik
Hi,

I've got a function that I'm trying to show is positively invarient.

$$Let \ V(x,y) = 0.45x^{2}+xy+0.56y^{2} > 0 \ for \ all \ x,y \ in \ R^{2}$$

$$V'(x,y) = ... = -8.4840x^{2} - 18.7412x - 10.3496 - 0.011xy - 0.011y^{2};$$

How can I find a neighborhood/region that makes V'(x,y)< 0?

Hi Somefantastik!

Complete the square (twice) … write it in the form -A(x - B)2 - C(x - D)2 +E

1. What is a negative definite 'hood?

A negative definite 'hood is a mathematical concept used in linear algebra and optimization. It refers to a set of points in a vector space that have negative values for all combinations of non-zero vectors. In simpler terms, it is a region in which all points have a negative curvature in all directions.

2. How is a negative definite 'hood different from a positive definite 'hood?

A positive definite 'hood is the opposite of a negative definite 'hood, where all points have a positive curvature in all directions. This means that a positive definite 'hood has a minimum value at every point, while a negative definite 'hood has a maximum value at every point.

3. What is the significance of finding a negative definite 'hood?

Finding a negative definite 'hood is important in optimization problems, as it helps identify the maximum value of a function. This can be useful in various applications such as machine learning, where maximizing a certain objective function is desired.

4. How is a negative definite 'hood identified?

A negative definite 'hood can be identified by determining the eigenvalues of the Hessian matrix of a function. If all eigenvalues are negative, then the function has a negative definite 'hood. Alternatively, the second derivative test can also be used to identify a negative definite 'hood.

5. Can a function have both a negative and positive definite 'hood?

Yes, a function can have both a negative and positive definite 'hood at different points. This means that the function has both a maximum and minimum value within its domain. However, a function cannot have a negative definite 'hood and a positive definite 'hood at the same point.

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