The art of persuading an audience through writing or speech is question 4 options: epideictic anachronism chiasmus rhetoric

Let's start by defining your four answer choices:

Epideictic - speech or writing used for ceremonies that praise or blame. An example of an epideictic speech would be one that a person delivers at a funeral, speaking about the life of the deceased.

Anachronism - using an object, event, custom, or person that is not in its correct historical time; sometimes on purpose and sometimes accidentally! Example: George Washington texting his whereabouts to Martha Washington on his iPhone X.

Chiasmus - two or more clauses or phrases are used, then reversed for effect. It's often used as a way to create memorable phrases. Songs, company slogans, commercials, speakers, and literature all use chiasmus. Example: "Ask not what your country can do for you - ask what you can do for your country." - JFK

Rhetoric - the art of speaking or writing effectively or persuasively; someone who is skilled in the use of speech. Those that write essays for major magazines or speeches for politicians use rhetoric. 

Rhetoric would be the best answer choice.