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.



Identify maxima for 2-Dimensional Function without knowing cross-derivative
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 $g\_{xy}=a$.


