I am trying to find the values of a and b for which the function $f(x,y)=x^a+y^b$ is strictly concave over non-negative x and y.
At first I was using the method of evaluating the definiteness of the Hessian. I got that the leading principal minor of order one is $a(a-1)x^{a-2}$ and the leading principal minor of order two is $a(a-1)b(b-1)x^{a-2}y^{b-2}$. Since the first must be negative and the second must be positive, this should require both $0<a<1$ and $0<b<1$.
But then I started wondering whether this solution is only valid for the domain where $x$ and $y$ are both strictly positive. What happens if $x=y=0$? How do I make an argument that $f(x,y)$ is strictly concave for $0<a<1$ and $0<b<1$ over its whole domain? The Hessian is not negative definite at $x=y=0$, but since negative definiteness is only a sufficient condition and not a necessary condition, that doesn't rule out strict concavity there, right? Should I somehow appeal to the definition of strict concavity?