Find Critical Points of Hessian Matrix

In summary, the critical points of the function f(x,y,z)=x^3+y^2+z^2+12xy+2z were found to be at (24,-144,-1) and (0,0,-1). The Hessian matrix was used to determine their types, with both being non-degenerate. The Morse index for the critical point at (24,-144,-1) is 2, while the Morse index for the critical point at (0,0,-1) is 1. The sign of the eigenvalues of the Hessian matrix can be used to determine the type of critical point (i.e. maximum, minimum, or saddle point), but further calculations are needed to determine the specific
  • #1
azatkgz
186
0
Please,check my solution.
Find critical points of the function [tex]f(x,y,z)=x^3+y^2+z^2+12xy+2z[/tex]
and determine their types (degenerate or non-degenerate, Morse index for non-
degenerate).

Attempt

[tex]\frac{df}{dx}=3x^2+12y=0[/tex]

[tex]\frac{df}{dy}=2y+12x=0[/tex]

[tex]\frac{df}{dz}=2z+2=0[/tex]

Critical points are at

x=24 y=-144 z=-1

x=0 y=0 z=-1

[tex]H(f)=\left|\begin{array}{l[cr]}6x&12&0\\12&2&0\\0&0&2\end{array}\right|[/tex]


for x=24

[tex]det\left|\begin{array}{l[cr]}144&12&0\\12&2&0\\0&0&2\end{array}\right|=288[/tex] non-degenerate


for x=0


[tex]det\left|\begin{array}{l[cr]}0&12&0\\12&2&0\\0&0&2\end{array}\right|=-288[/tex] non-degenerate
 
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  • #2
azatkgz said:
Please,check my solution.
Find critical points of the function [tex]f(x,y,z)=x^3+y^2+z^2+12xy+2z[/tex]
and determine their types (degenerate or non-degenerate, Morse index for non-
degenerate).

Attempt

[tex]\frac{df}{dx}=3x^2+12y=0[/tex]

[tex]\frac{df}{dy}=2y+12x=0[/tex]

[tex]\frac{df}{dz}=2z+2=0[/tex]

Critical points are at

x=24 y=-144 z=-1

x=0 y=0 z=-1

[tex]H(f)=\left|\begin{array}{l[cr]}6x&12&0\\12&2&0\\0&0&2\end{array}\right|[/tex]


for x=24

[tex]det\left|\begin{array}{l[cr]}144&12&0\\12&2&0\\0&0&2\end{array}\right|=288[/tex] non-degenerate


for x=0


[tex]det\left|\begin{array}{l[cr]}0&12&0\\12&2&0\\0&0&2\end{array}\right|=-288[/tex] non-degenerate

keep going... are they minimum, maximum.. saddle points??
 
  • #3
Morse index

for (0,0,-1)

[tex]det\left|\begin{array}{l[cr]}-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0[/tex]

[tex](2-\lambda )(-\lambda(2-\lambda)-144)=0[/tex]

[tex]\lambda_1=2,\lambda_2=1-\sqrt{145},\lambda_3=1+\sqrt{145}[/tex]


for (24,-144,-1)



[tex]det\left|\begin{array}{l[cr]}144-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0[/tex]

[tex](2-\lambda )((144-\lambda)(2-\lambda)-144)=0[/tex]

[tex]\lambda_1=2,\lambda_2=73-\sqrt{5185},\lambda_3=73+\sqrt{5185}[/tex]

Is it right?What we can say about maximum,minimum and saddle points?
 
  • #4
azatkgz said:
Morse index

for (0,0,-1)

[tex]det\left|\begin{array}{l[cr]}-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0[/tex]

[tex](2-\lambda )(-\lambda(2-\lambda)-144)=0[/tex]

[tex]\lambda_1=2,\lambda_2=1-\sqrt{145},\lambda_3=1+\sqrt{145}[/tex]


for (24,-144,-1)



[tex]det\left|\begin{array}{l[cr]}144-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0[/tex]

[tex](2-\lambda )((144-\lambda)(2-\lambda)-144)=0[/tex]

[tex]\lambda_1=2,\lambda_2=73-\sqrt{5185},\lambda_3=73+\sqrt{5185}[/tex]

Is it right?What we can say about maximum,minimum and saddle points?

i didnt check you're calculus, but find what the sign of eigenvalues mean and you'll get you're answer.

bye
 

Related to Find Critical Points of Hessian Matrix

1. What is the Hessian Matrix?

The Hessian Matrix is a square matrix of second-order partial derivatives of a multivariable function. It is used to determine the nature of critical points, including maxima, minima, and saddle points, of a given function.

2. How do you find critical points of the Hessian Matrix?

To find critical points of the Hessian Matrix, you must first calculate the gradient vector of the function and then find the second partial derivatives with respect to each variable. Next, you will plug in the critical points into the Hessian Matrix and determine the nature of the critical points by examining the signs of the eigenvalues.

3. Why do we need to find critical points of the Hessian Matrix?

Finding critical points of the Hessian Matrix allows us to identify the maxima, minima, and saddle points of a multivariable function. This information is useful in optimization problems and can help us understand the behavior of a function in a given region.

4. Are there any shortcuts for finding critical points of the Hessian Matrix?

Yes, there are some shortcuts that can be used to find critical points of the Hessian Matrix. For example, if the Hessian Matrix is diagonal, the eigenvalues are simply the second partial derivatives, making it easier to determine the nature of the critical points. Additionally, for certain types of functions, such as quadratic functions, the critical points can be found using algebraic methods without having to use the Hessian Matrix.

5. Can the Hessian Matrix be used for functions with more than two variables?

Yes, the Hessian Matrix can be used for functions with any number of variables. However, as the number of variables increases, the calculation of the Hessian Matrix becomes more complex and may require advanced techniques such as the Jacobian matrix. Additionally, interpreting the results and determining the nature of the critical points becomes more challenging as the number of variables increases.

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