Hessian matrix in taylor expansion help

[sdevoe]

Homework Statement

Find the critical point(s) of this function and determine if the function has a maxi-
mum/minimum/neither at the critical point(s) (semi colons start a new row in the matrix)

f(x,y,z) = 1/2 [ x y z ] [3 1 0; 1 4 -1; 0 -1 2] [x;y;z]

Homework Equations

The Attempt at a Solution

I'm fairly certain this is the second derivative of a taylor series expansion so 3rd term. So the matrix [3 1 0; 1 4 -1; 0 -1 2] is the Hessian. What I do not know now is how to get the maximum/minimum/neither or the critical points from the hessian.

 

[sunjin09]

The critical points are such that all the partial derivatives are 0

 

[sdevoe]

So does that mean where 3x+y=0, x+4y-z=0, and -y+2z=0?

 

[sunjin09]

So does that mean where 3x+y=0, x+4y-z=0, and -y+2z=0?

yes you are right

 

[sdevoe]

Confirming that I have to solve it as a system of equations?

 

[Ray Vickson]

Confirming that I have to solve it as a system of equations?

Yes, of course. That is exactly how critical points are found, in general.

RGV

 

[sdevoe]

I will get that all the values are equal to zero if I solve that?

 

[Ray Vickson]

I will get that all the values are equal to zero if I solve that?

Try it and see.

RGV