# Critical points of function in 3 variables

1. Aug 1, 2010

### K29

1. The problem statement, all variables and given/known data
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.

Last edited: Aug 1, 2010
2. Aug 1, 2010

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.

3. Aug 1, 2010

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.

4. Aug 1, 2010

### K29

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