Which one state is the farthest east, west, north, AND south in the US? This is just one state for each direction.
There must be something slightly wrong with the question as it was given to you. There is no one state that can be seen as being the farthest east, west, north and south in the United States. There is a way to say that one state is the farthest east, west, and north, but not south. We can say that Alaska has the points in the US that are farthest in three directions, but there is no way that it is farthest south.
Alaska is clearly the state in the US that has the northernmost point. Alaska is well north of any state in the contiguous 48 states. Alaska also has the points that are farthest west and east, at least in one way of looking at things. This is because the Aleutian Islands extend past the Prime Meridian. This means that some of those islands are in the Eastern Hemisphere and are therefore technically east of anything else in the US. However, there is no way to argue that Alaska has the southernmost point in the United States. That point is in Hawaii.
I concur with the first answer. I think that the question must be asking, "Which state extends the farthest in overall area to the north, south, east and west. That would pretty obviously be the state of Texas. I'm not sure I make myself clear. It might be better to phrase it as: Which state has the widest length in area from east to west and the greatest length in area from north to south?