Identify maxima for 2-Dimensional Function without knowing cross-derivative

Question

I am trying to proof the uniqueness of a maximum for a two-dimensional function (well behaved, twice differentiable, domain $R^2$, etc.), yet cannot compute the exact derivatives or the Hessian.

I have $f(x,y) = g(x,y) - bx - cy$ and know that $g_{x}>0, \ g_{y}>0, \ g_{xx}<0$ and $g_{yy}<0$, but do not know $g_{xy}$. Also, $b>0$ and $c>0$.

Is that sufficient structure to say anything about the existence of single/multiple maxima? $g(x,y)$ is a function without close-form solutions, but if there was a particular property missing to claim uniqueness, I could try and proof that too.

Any help is much appreciated - if you have any pointers to relevant text-books or papers, please let me know. So far I was using Sundaram's book on Optimization Theory.

Answer 1

Of course, nothing definite can be said here. E.g., let $b=c=1$ and $g(x,y)=x+y-x^2-y^2+axy$ for some real $a$ and all real $x,y$, so that $f(x,y)=-x^2-y^2+axy$.

Then, if $|a|<2$, then $(0,0)$ is the only point of (local and global) maximum of $f$. If $|a|>2$, then $(0,0)$ is a saddle point of $f$ and there is no point of local or global maximum of $f$. If $|a|=2$, then there are infinitely many points of local (and global) maximum of $f$. So, here everything hinges on $g_{xy}=a$.