If you take x< y, second function is always increasing as x increases as it becomes 1/2 f(x,x). F(x,x) is increasing function.
If you take x >y, then first term is increasing for increase in x. It becomes f(x,y) - 1/2 f(y,y). But for increase in y we are not sure.
When y increases, second function remains constant. First function can decrease or increase when y increase, as first term and second term both are increasing.