Sentence structure of "At what age"
Leaving aside the problem of whose age (the addressee's or the baby's) is being questioned,
- At what age were you able to start reading successfully to a baby?
is an example of a grammatical phenomenon known as Pied-Piping. This sentence is a question, formed from a statement something like
- You were able to start reading successfully to a baby at
___age
where the blank represents the missing information the questioner is seeking; in this case it gets transformed into the interrogative what in the prepositional phrase at what age.
- You were able to start reading successfully to a baby at what age
The next step in question formation is to invert (exchange) the subject (you) and the first auxiliary verb (were)
- Were you able to start reading successfully to a baby at what age
If it weren't for the Wh-word at the end, this would be an OK yes/no question
- Were you able to start reading successfully to a baby at that age?
But it's a Wh-question, not a yes/no question, so after inverting, the next step is to move the Wh-word (in this case, the phrase what age) to the front of the question, producing
- What age were you able to start reading successfully to a baby at?
Which is grammatical, but is also a great example of stranding a preposition at the end of the sentence. What good is it doing there? It doesn't mean anything; it's just something required by the idiom be at X age, and it's silly to require the speaker to add it at the end, or for the listener to have to wait for it. But you can't leave it out. Because grammar.
So, the optional rule of Pied-Piping allows the speaker to move the whole prepositional phrase at what age, as a unit interrogative constituent.
Not only that, but any other higher constituent that the interrogative word is part of -- like a noun phrase containing the prepositional phrase, or another noun phrase containing that one -- can also be pied-piped, at the will of the speaker, producing prose of massive turgidity, like
- printed volumes the lettering on the covers of which is specified by law