How do I reach the students who have checked out completely?

You can hold most of the room. It is the ones who have gone quiet inside, or who push back on everything, that wear you down. Those students feeling impossible is not a mark against your teaching. They are simply the hardest ones, and you have already reached everyone easier. A checked-out or resistant student is almost never bored by you personally. Usually they are protecting themselves from something: a fear of failing, a history of being made to feel stupid, a wall built long before they entered your class. The resistance is armor, not a verdict. Your task is not to break it but to make it unnecessary. Pick one of these students and, away from the lesson, show them one small sign that you see them as a person and not a problem. Ask about something they care about. Notice one true thing they did well. You may not win them today. But most walls come down slowly, from the safe side, and yours may be the first hand that does not push.