Critical points of function in 3 variables

[K29]

Homework Statement

I have the scalar function
$$T(r)= 2x^{2} - 4x +y^2 +4y-3z^{2}$$ (r is obviously a vector)
I have the critical point (1,-2,0) but I'm not sure how to work out if its a min/max/saddle. I'm familiar with doing it for functions in 2 variables.

 

[Raskolnikov]

No, for functions of 3+ variables, you have to look at the eigenvalues of the Hessian matrix at the critical points.

If all eigenvalues are positive, then crit. pt. is a minimum. If all negative, then crit. pt. is a maximum. If some of the eigenvalues are negative while the rest are positive, then the crit. pt. is a saddle point. If any of the eigenvalues are equal to zero, then our test is inconclusive.

 

[Raskolnikov]

Oh, and the Hessian matrix is the nxn matrix of 2nd partial deriv's of your function, where n is the number of variables (3 in your case). Look it up on wiki if you're unfamiliar.

 

[K29]

Ok thanks. I edited the question while you were replying, but I understand. To anyone else this will look a mess.