Using the Hessian to determine convexity of $f(x,y) = \ln(x + y)$

Question

Disclaimer: This is a homework question that I'm currently working on.

I have been asked to use the Hessian to determine if $f(y_1, y_2) = \ln(y_1 + y_2)$ is convex, concave, or neither.

I've computed the hessian as $$H_f(y_1, y_2) = \frac{-1}{(y_1 + y_2)^2} = \begin{bmatrix} 1 & 1 \\ 1 & 1\end{bmatrix}$$

$H_f$ is singular ($\det(H_f) = 0$) everywhere. So I am tempted to conclude that this function is neither concave nor convex.

However, I also know that the composition of two concave functions is concave. And I believe that $f_1(x,y) = x + y$ and $f_2(x) = \ln(x)$ are both concave.

What am I missing here?

Answer 1

$\det(H_f)=0$ yes but what is equivalent to $f$ concave is $H_f$ being semidefinite negative, which is the case here.