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why are different methods giving different answers?

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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.
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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?
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