Is it permissible for him to take from the produce of a palm tree he inherited solely from his father, bearing its full costs due to his brothers' refusal to participate, and their silence regarding his actions?

What the deceased left behind, such as a farm and other assets, is the right of all heirs, and it is not permissible to dispose of it except with their permission.

If the adult siblings remain silent and do not object to watering the farm, tending to it, and taking the harvest, their silence is considered permission to dispose of their share, unless it is highly probable that their silence is out of shyness, especially if they are women; because "the property of a person is not permissible except with their willing consent."

However, if there are minor, non-adult heirs among the inheritors, their silence or permission is not considered. If their guardian gives you permission, you are considered a hired worker for their share, and you take your wage after selling the harvest, returning the remainder to them. The wage must be determined beforehand. As for what has passed, you are entitled to the wage customary for such work.

The advice is to speak with your siblings and obtain explicit permission from the adult ones among them, and not to suffice with mere silence.