elicit
To carefully draw out a reaction, answer, or information.
To elicit a response or reaction means to draw it out or bring it forth, often through careful questions or actions. When a teacher asks thought-provoking questions to elicit discussion from the class, she's trying to get students talking and sharing their ideas. A scientist might design an experiment to elicit a specific reaction from chemicals. A comedian tells jokes to elicit laughter from the audience.
The word suggests some skill or effort in getting the response. You don't elicit something that happens automatically. Instead, you elicit information that someone might not volunteer on their own, or reactions that need the right conditions to appear. A detective might elicit a confession by asking questions in just the right way, or a researcher might elicit memories by showing people old photographs.
Usage note: Don't confuse elicit with illicit, which means illegal or forbidden. They sound similar but mean completely different things. You might elicit the truth (draw it out), but you wouldn't want to do anything illicit (against the law).