Consider $f:\mathbb{R}^{2}_{0} \to \mathbb{R}_{0}$ such that $f(x,y)$ is a continuous function and satisfies the following properties:

1. $f(x,y) = f(y,x)$
2. $f(tx,ty) = tf(x,y) \ \forall \ t > 0 $
3. $f(1,1) = 1$

Can we show that if $g(x,y) := 3f(x,y) - 2(x+y)$, then $\underset{x,y}{\text{argmax}}[g(x,y)] = (0,0)$ assuming a maximizer exists?

I can only show that $g(x^{*},y^{*}) =0$ since otherwise $g(2x^{*},2y^{*}) > g(x,y)$ yields a contradiction if $g(x^{*}, y^{*}) \neq 0$. I can neither think of a counter-example nor a proof to complete the solution.

Note that $f$ is not necessarily (partially) differentiable. $\mathbb{R}_0$ is the set of non-negative reals.