# Homework Help: 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