I assume you are talking about using all three words as a verb. The words "try" and "test" can also be used as nouns, but "prove" can only be a verb (the noun form of "to prove" is "proof"). "To prove" (verb) means to demonstrate (or to show) that something is true, and implies that evidence must is needed. Examples: I will show you a map and PROVE that we have traveled five miles. Can you PROVE that your story is true? A very common exchange in American English: Person A makes a statement, for example, "My dog is smarter than your dog." Person B challenges with, "Prove it." "To try" (verb) means to make an effort to do something. Examples: I will try to do well on the test. Please try to fix my car engine. A common exchange in American English: Person A, explaining why he has failed, may say, "I tried my best." (meaning: I gave it my best effort.) Person B may reply, "Try harder." (meaning: you should be able to make a better effort, so make a better effort and succeed.) "To test" means to check the ability or quality of something, perhaps to determine how fast a car can go, or how much a person knows. Examples: We will test the new race car on the track tomorrow. I may test your knowledge of history.

Try and test are some how the same you can use them both in a sentence. let's try it! let's test it! they both have the same meaning. test: 1.use or examine sth carefully to find out if it's working properly or what it is like: these cars have all been tested for safety. 2.a short medical examination of a part of yor body: to have an eye test. try: 1. to make an effort to do sth: I tried to phone you but I couldn't get through. 2.try sth: to do , use or test sth in order to see how good or successful it is: we tried the door but it was locked. but prove has a totally different meaning.--> prove sth: to use facts and evidence to show that sth is true: it will be difficult to prove that she was lying. she tried to prove her innocence to the court.