I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$).
Plots of the function indicate it to be concave. Both the numerator and the denominator are convex and positive. But approaching $f(x,y)$ as a ratio of two convex functions (with different gradients) didn't help in my attempts towards a concavity proof. Useful suggestions would be highly appreciated.
Here is a plot of the function:
