# Finding extrema of function of three variables

So i have to study this function and find out if there are any local or absolute extrema :

$f:\mathbb{R}^3 \rightarrow \mathbb{R} :$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2 + \left(z-\sqrt{x^2+y^2}\right)^3$$

I've read that, for 3 or more variable functions, if the Hessian is positive definite at a certain point then f has a minimum and if it's negative it has a maximum, but how do i see if it's positive/negative definite at that point?

• Because your Hessian will be a Hermitian matrix you can use: en.wikipedia.org/wiki/Sylvester%27s_criterion – A. A. Jul 27 '15 at 15:15
• have you get the derivatives? – Dr. Sonnhard Graubner Jul 27 '15 at 15:30
• @Adolfo Do you know if there is any other way of solving this problem without using matrices? And thanks for the help! – J. Barbosa Jul 27 '15 at 17:01
• @JoãoBarbosa but you know for a continuous function the hessian matrix is always a hermitian one and being positive-definite or negative-definite is just defined for hermitian matrices. so in order to study about the extremas of a multi-variable function you should be familiar alittle with hermitian matrices. you don't need to know every thing about them but you should be able to say that whether a hermitian matrix is positive definite or negative definite. It's not a very difficult concept maybe the page Sylvester's criterion confused u – Sepideh Abadpour Jul 27 '15 at 22:23
• @sepideh ye the criterion is relatively easy to apply and i think i got it down! Thanks for all the help! – J. Barbosa Jul 28 '15 at 17:05