Critical Point or Saddle Point $3$ variable function

Question

I am given a $3$ variable function:

$$f(x,y,z) = \cos(xy) + \cos(yz) + \cos(zx)$$

I found that there is a critical point at the origin by taking the partial derivative with respect to $x$, $y$, and $z$, and found a critical point at $(0,0,0)$. However, now I am confused because I don't know how to do the Hessian matrix for a $3$ variable function.

Answer 1

Notice that since $\cos(w) \le 1$, hence $f(x,y,z) \le 3$.

Also note that $f(0,0,0) = 3$, hence $(0,0,0)$ must be a global maximum.

No computation of hessian is needed.

Remark:

The hessian is the matrix $\begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \\ f_{yx} & f_{yy} & f_{yz} \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}$