The correct answer is true.
It is true that the first complete cookbook in western civilization was written by Apicius in the first century AD.
We are talking about the Roman citizen Marcus Gavius Apicius. Apicius was a refined Roman man that liked to enjoy a good and luxurious life. He enjoyed preparing fish and seafood and created some recipes with different ways of preparing food. Although cannot be confirmed by historians, it is believed that he wrote the Roman cookbook.