Finding maximum of function with two variables

I have question about finding maximum of function in two variables, the specifics are:

$$V(a,b)=\frac{1}{r}\Big[a(e^{r(1-a-c_b b)})-r(1-a-c_b b)\Big]+\frac{1}{s}(2-c_b)b(e^{s(1-b-c_a a)}-1),$$where $a,b\in[0,+\infty)\times[0,+\infty)$ are variables, and $r,s,c_b\in(0,1)$ and $c_a>1$ are parameters. Task is to prove, that $V(a,b)<0$ everywhere, up to a point $(a,b)=[1,0]$ in which holds $V(a,b)=0$.

Idea was to prove that point [1,0] is global maximum, but probably is better way, we just need to prove that $V(a,b)<0$ everywhere, and at point [1,0] holds $V(1,0)=0$. It is quite hard or impossible to find all stacionary states, so this task requires specific approach.

If we draw graph of this function it is obvius that $[1,0]$ is global maximum, but we need to show it analyticaly.

Thanks for suggestions and ideas.