Both are fine.
Which is correct?
1) What do you think is the purpose of the article?
2) What do you think the purpose of the article is?
Both are fine.
As far as I know, there is no subject-verb inversion when "do you think" is embedded in a question. But I did find the #1 is used in a textbook. So I am wondering why.
***** NOT A TEACHER *****
(1) May I add my two cents?
(2) Mr. Henry Fowler wrote a book entitled A Dictionary of Modern English Usage. I have the 1965 edition. Some people feel it's one of the best books on correct English; others feel otherwise. Anyway, Mr. Fowler (IF I have understood what I read) gives a brilliant explanation for sentences similar to yours.
(3) Mr. Fowler explains why this particular kind of sentence is an exception. [My note: Of course, one must ask:
"Can you tell me what time it is?" It's "bad" English to say "Can you tell me what time is it?]
(4) Let me try to tell you what the great Henry Fowler said about some similar sentences. You can then apply the ideas to your sentences.
(5) Mr. Fowler says that if you say "Explain what are the duties," then the answer would be "These are the duties." *** If you say "Explain what the duties are," then the answer would be "The duties are these."
(6) He says both indirect questions are "normal." He emphasizes that this kind of
subject + be + complement question ("What are the duties?") is a "minor" point.
(7) In fact, it is so minor (unimportant) that the 1996 edition of his book has deleted this material. (I'm lucky to have the 1965 edition; the master wrote the first edition in 1926.)
(8) I hope that you can use Mr. Fowler's reasoning to figure out why both of your
sentences -- as the moderator told us -- are "fine."
Your(TheParser's) reply seems to suggest that In "What is the purpose of this article?", "what" can be either the subject or the complement. Generally speaking, "what" seems to be considered the subject in the sentence. So when someone asks "What(Subject) is the purpose of the article?", the answer is usually "The purpose of the article is to exaplain the sport of Formula 1 car racing." Here "what" is used as the subject. But it also seems possile that we can use "what" as a complement in such a sentence as this:
What(Complement) is the purpose of this article?
To explain the sport of Formula 1 car racing is the purpose of the article.
Although both the sentences may be correct, I think that the first one ("what" used as the subject) is a preferred form for many people.