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He is asked to investigate [what the police believe to be the root of the crime (to discover the truth.)]

The first infinitive to be the root of the crime, the understood subject is what, right?
Can the noun phrase be viewed as: The police believe what to be the root of the crime?

The second infinitive to discover the truth is a purpose adjunct,
and the understood subject is he,correct?

He became what his [friends wouldn't expect him to be] (to prove them wrong.)

the understood subject of the first infinitive to be is him , and the object of be is what , right?
Can the noun phrase be viewed as: His friends wouldn't expect him to be what.

And for the second infinitive, to prove them wrong is a purpose adjunct , and the understood subject is He,correct ?

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Yes, your analysis is right.

Your paraphrases aren't actually grammatical, because you can't use what in that way (unless it is interrogative, which isn't the case here). But the understanding you have used to create them is right.

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