A following function is given: $$ f(x,y,z)= x^2 + \frac{2}{x} + (2z+y)^2 + y^2 + \frac{2}{2z+y} + \frac{2}{y} $$
I know, that i have to start by calculating partial derivatives in respect to x, y and z, and this is easy for me. But I don't really comprehend what I should do next. I'd appreciate it if someone could show me how to appropriately solve this kind of problems.
Then you determine at which points the gradient is $0$; there is only one such point, which is $(1,1,0)$. And then you compute the Hessian of $f$ at that point, in order to try to determine whether it is a local maximum or a local minimum.