Infinitives can be identified by the presence of "to" after some verbs. They have a very specific function and mostly appear after the action, which in the sentence is "hoping". The right answer is the third one: an infinitive functioning as an adverb. A verbal seems to be a verb, but its function is to complement a verb and plays the role of a different part of speech. "To begin" is an adverb because it modifies the action of hoping, it gives additional information of what is being expressed.