Question

why are different methods giving different answers?

Answer 1

I think both of your conclusions are wrong, you can't determine if the function is decreasing or increasing from the given information. f = exp and f = log are examples where g(x) = f(x) + f(2-x) is increasing and decreasing on (1,2) respectively.

Answer 2

Yes f’(x) > 0, that was given. So f(x) is increasing. How are you going from that to f(x) - f(2-x) is increasing?