Let $$a(x)=\alpha f(x)+\beta+g(x) $$ and $$b(x)=\gamma f(x)+\delta +g(x)$$where $f,g$ are continuous real-valued functions on a closed interval $[0,c]$ and $\alpha,\beta,\gamma,\delta$ are constants with $\alpha,\beta\geq0$.
Is it possible to prove that: $$\arg\max_{x\in[0,c]}a(x) = \arg\max_{x\in[0,c]}b(x) $$And if so, under which conditions (for instance $\gamma>0$) ?
Assume for a moment that your functions are differentiable, then necessary condition for optimality would give
$$0=\alpha f'(x)+g'(x),$$
$$0=\gamma f'(x)+g'(x).$$
In the general case, there would be no reason why the solution to these equations would be the same.
Otherwise, you can simply pick for example $g(x) = -x^2$ and $f(x) = M \geq 0$.