Choose the correct form of the following word, out of four given options, as…

Concept: a word’s part of speech is fixed by the job it does inside its own sentence, not by its spelling, and the word "well" can take three different jobs. As an adverb it attaches to a verb, a participle, an adjective or another adverb and answers the question "how?". As a predicative adjective it follows a linking verb such as "be", "seem", "look" or "feel" and describes the subject, usually with the sense "in good health". As a noun it names a state or a thing and fills a subject or object slot.

Application: in "Well begun is half done", "well" stands immediately before the past participle "begun" and answers "begun how?". Attaching to a participle in order to state the manner of the action is exactly an adverb’s job, so this is the sentence in which "well" is used as an adverb.

• "Let well alone" — here "well" follows the verb "let" and fills the object slot, naming the satisfactory situation that is to be left untouched, which is a noun use of the word.

• "I hope you are now well" — here "well" follows the linking verb "are" and describes the subject "you" with the sense "in good health", which is the predicative adjective use.

• "With his name the mothers still their babies" — the word "well" does not occur in this sentence at all; its verb is "still", used in the sense "to quieten".

Cross-check: substitute a plain adverb of manner and see which sentence survives. "Nicely begun is half done" still reads naturally, which confirms the adverbial slot, while "I hope you are now nicely" and "Let nicely alone" both break, which confirms that those two uses of "well" are not adverbial.