TITLE: Critical Point or Saddle Point $3$ variable function
QUESTION [0 upvotes]: 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.

REPLY [0 votes]: 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 $\begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \\ f_{yx} & f_{yy} & f_{yz} \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}$
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