Explanation

The first and third blanks are highly related, so consider them first. You learn in the first clause of the sentence that there is “antagonism,” or hostility, between two people, but from the word apparent you learn that there is more to this than meets the eye.

More than pure “hatred,” or antipathy, is something that allows the two individuals to “derive inspiration and motivation from the other.” In other words, there is a mutual benefit to the seemingly hostile relationship.

Now return to the second blank. Because the structure of the sentence tells you that the antagonism could be written off, but further scrutiny renders it (blank), you know you want a word that goes in the same direction (that is, you will not be “writing off” the antagonism). The best word here is patent, which, when used as an adjective, means “obvious” or “evident.”

Antipathy, patent, mutual benefit.