The correct answer is option 3: The Narrow Road to the Deep North.

Let`s analyze each option to understand why the correct answer is option 3.

Option 1: History of the Rain. This book was not the winner of the Man Booker Prize in 2014.

Option 2: The Lives of Others. This book was not the winner of the Man Booker Prize in 2014.

Option 3: The Narrow Road to the Deep North. This book was indeed the winner of the Man Booker Prize in 2014.

Option 4: How to be Both. This book was not the winner of the Man Booker Prize in 2014.

Therefore, the correct answer is option 3: The Narrow Road to the Deep North. It is important to note that the Man Booker Prize is a prestigious literary award, and winning it is considered a significant achievement for an author.