Which word means to give a false appearance of something?

The word "feign" is indeed the correct choice because it specifically refers to the act of pretending or giving a false appearance of something. This term is often used in contexts where someone is not being genuine, such as feigning interest or feigning an illness. By creating a facade or a misleading impression, the term captures the essence of deception or masquerading as something that one is not.

The other options, while they have distinct meanings, do not convey the same concept of creating a false appearance. For example, "exploit" refers to taking advantage of a situation or person, often in a context of selfish gain. "Falter" means to hesitate or waver in action, which does not align with the idea of deception. "Extinguish" refers to putting an end to something, such as a fire or a feeling, and similarly does not connect to the notion of pretending or creating a false front.