These words are all very close in meaning and often interchanged. Below is just how I understood their subtle differences after looking them up in a dictionary. I think this is a very difficult question, so I hope to see other people's input. They are all a type of MENTAL DISCERNMENT/UNDERSTANDING THROUGH FOCUSED OBSERVATION. Astute has a bit of a subtle overtone of a desire to GAIN ADVANTAGE. Someone who can assess a situation or a person, then use what he has perceived for his advantage. But it carries a more positive meaning. An astute businessman is someone who is good at observing what is good for his business, not someone who is negligent of other people's rights for his own advantage. Perspicacious has a subtle overtone of WISDOM and EXPERIENCE. A perspicacious judgement is one made with a clear head and ready knowledge. Perceptive has a subtle overtone of INTUITION. A perceptive student is able to see something and think about it clearly, to understand it. This doesn't mean he will have some sort of gain from this or that he has wisdom. The student is just intelligent and has a good ability to grasp something through his sense of mental observation.

A dictionary that I would recommend to you is: www.thefreedictionary.com

to me, if someone were considered perspicacious, they would be expected to have great deductive and logical reasoning skills, finding out important facts by logical deductions--Sherlock Holmes is an excellent example of perspicious (note, Perspicacity (noun) is related here) Astute--to me has a greater meaning of praising one's general intelligence and intellectual abilities, i.e. good students are often said to be "astute learners" Perceptive to me refers to someone who pays attention to every detail in their environment. Importantly, although these words are distinct in their implications, they are often used interchangeably when such implications are not intended or important, although I would say that perspicacious is certainly found more in formal writing than in everyday conversation. astute and perceptive in particular could be used to mean the same thing. Logically, they words at a deep level carry subtle differences and are independent. For example, if someone is perspicacious, I would expect them to also be astute and perceptive, though it is possible for an astute person not to be perspicacious, i.e. not to reach the probable conclusions from deduction.