The correct answer is option 2 - Dhauli major rock edicts.

Before explaining the correct answer, let`s briefly understand what an inscription is. An inscription is a piece of writing or marking on a surface, typically made on a stone or metal, that provides information or records an event. In the case of Ashoka, these inscriptions were important in conveying his policies and edicts to the people.

Option 1 - Barabar hill cave inscription is not the correct answer because it refers to inscriptions found in the Barabar hills of Bihar, which are not in the easternmost part of Ashoka`s territory.

Option 3 - Sahasram minor rock edicts is also not the correct answer because this refers to a different set of inscriptions found in Odisha, which are located more towards the eastern-central part of Ashoka`s territory.

Option 4 - Lauriya Araraj pillar edicts is not the correct answer either. These inscriptions are located in present-day Bihar, which is geographically closer to the central part of Ashoka`s territory.

Therefore, the correct answer is option 2 - Dhauli major rock edicts. These inscriptions were installed in Dhauli, Odisha, which is in the easternmost part