Is the function convex or concave?

Question

I am trying to solve question below: Is function $$f(x, y, z) = x+y +z + \ln(xyz)$$ strongly convex or strongly concave in the $A = \{ (x, y, z) | x > 0, y < 0, z < 0 \}$? Also, determine the (possible) local and global extremes values?

Answer 1

You want to evaluate the signature of $Hf$ in the set of interest. On $A$, $$f(x,y,z)=x+y+z+\ln x+\ln(-y)+\ln(-z)$$ and therefore $$Hf(x,y,z)=\begin{pmatrix}-x^{-2}&0&0\\ 0&-y^{-2}& 0\\ 0&0& -z^{-2}\end{pmatrix}$$

Therefore the function is concave on $A$.

The critical points are solutions in $A$ to $\nabla f(x,y,z)=0$, i.e. $$\begin{cases}x>0\\ y<0\\ z<0\\ 1+\frac1x=0\\ 1+\frac1y=0\\ 1+\frac1z=0\end{cases}$$

So no critical points, meaning that the function has no local maxima or minima. It's clear that $\sup_{v\in A}f(v)=\infty$ and $\inf_{v\in A}f(v)=-\infty$.