We tend to think of "vulgar" in the sense of "vulgarity" or the use of swear words; however, this notion comes from the general notion or meaning of "vulgar," which comes from the Latin meaning pertaining to the ordinary people. Something vulgar is lacking in refinement or culture. Therefore, the best antonym among the options provided is "refined," meaning elegant or cultured.

When an action is "blatant," it is done without any attempt to hide it. Perhaps you have heard the expression used, "That was a blatant lie," (or some other close expression like that). Such a sentence means "That was a total lie—without any attempt to hide the fact." In contrast to this, a concealed act is one that is hidden—or at least an attempt is made to hide the act. This is a good contrast to something done blatantly.

The verb "inoculate" is likely more familiar than the adjective "innocuous." When we get inoculated for a virus, we receive a vaccine (or other kind of treatment) to remove the danger of getting the illness in question. Something that is "innocuous" is therefore something that is not dangerous or harmful. When something is "detrimental," it causes harm. Often, we use this word in expressions like "It was much to his detriment that he did not study a language at a younger age." The idea in such expressions is that it harms his present state of life that he did not study a language when he was younger. Something detrimental is not innocuous.

The word "brisk" can refer to cool weather, but it can also be used as an adjective describing fast movement. The adverb "briskly" means quickly or with energy. A brisk motion is one that is fast, and walking briskly means walking quickly with haste. In contrast to this, a leisurely pace is one that is slow—taking its time. Leisure is the free time used for enjoyable activities. Such time is not usually stressful. Therefore, "leisurely" activities are quite slow in their pace.

We are looking for something that simply implies irregularity as opposed to regularity. When something is done inconsistently, it is done without a regular manner or mode. A consistent tempo is one that is regular and repeated. Therefore, "inconsistently" is a good antonym for "regularly."

To be "coarse" means to lack refinement or to be "rough." Indeed, the word first has the physical sense of being rough—like a coarse, rough piece of sandpaper. In contrast to this, something that is delicate is fine in quality. Thus, to do something "delicately" is to do that thing in a way that is opposite to being "coarsely" done.

The adverb "fortunately" describes a circumstance that is positive or happy. To say, "Fortunately, a student raised her hand in response to the question," is like saying, "It is a fortunate and positive thing that a student raised her hand in response to the question." The word "lament" means to wail or otherwise express sadness greatly. When we say that something "lamentably happened," we mean to say that it is unfortunate or sad that it happened. This is quite the opposite of something fortunately happening.

A number of these wrong answers are trying to appeal to biases that have been created by stereotypes in our culture regarding being politically liberal. However, such stereotypes are unadvisable—both in day to day life and when answering a question like this one. Therefore, we must look for another answer. The adjective "liberal" means generous when used in a phrase like "a liberal amount of water." The adverb "barely" is a good contrast to "liberally"—think of "barely enough water" and "liberally poured water."

When something is solemn, it is a serious affair. A "solemn" church service is one that is weighty and given importance. A "solemn" speech is one that is given on an important day or about an important matter. To do something "solemnly" is to do it in a way that recognizes such importance. In contrast to this, to do something "frivolously" is to treat it as being unimportant. This is derived from the adjective "frivolous," meaning not having a serious value. It is also related to "frivolity," meaning light-hearted activity without any real weight.

The adjective "cheerful" is, of course, very familiar. It generally is taken to mean "happy." In addition to this basic sense, both the adjective and the adverb "cheerfully" connote a kind of optimism as well—i.e. looking on the "bright side" of things. Pessimism or hopelessness is opposed to such cheerfulness; therefore, the best option among those given is the adverb "hopelessly."