Pleasing two groups that start with contradictory opinions requires persuading at least one group to change it's mind. That's definitely possible because if they contradict then at least one of them is mistaken and would benefit from changing its mind.
Let's assume there exist two groups of people. The members of group A think that some improved product is better because it tastes great. The members of group B, on the other hand, believe that the same product is better because it's now less filling. Keeping in mind that the product is better because it both tastes great, and is less filling. Which group is mistaken? Which group would benefit from changing their minds?
Their ideas don't really contradict. But they think they do because, apparently, both sides have the mistaken view that a product has to be better for exactly one reason.
My example was obviously a gross simplification, but not all disagreement stems from contradiction. Worse, sometimes when people agree on many points they are in inreconcilable disagreement with each other on others. If we were to consider that both groups have held their position for many years, it should be immediately apparent that it would be impossible for either to change their mind. Time calcifies opinion.
You are using the word "impossible" to mean "very hard", and I am not.
You're right, though I think I would say "extremely difficult, tending toward impossible". People do not change their minds easily, especially on subjects they feel a strong emotional attachment to.
The people you're trying to please have contradictory opinions and desires. What you say to please Group A will strike Group B as an outrage! And if you try to remove everything potentially offensive to anybody, well, that will turn you into such a pandering namby-pamb that the whole world will find you annoying and want to poke you with a stick.
Edit: Just to be clear, I'm not saying one should go around offending people. I'm saying that trying to get everyone to like you is hopeless and that effort put into doing so will both warp and weaken you.
Other edit: just noticed that pchristensen already made the same point.