I watched an ICTP lecture on elementary real analysis & the lecturer went to great pains to emphasize the importance of the intermediate value theorem because it is what generalizes to higher dimensions via connectedness & how Bolzano-Weierstrass generalizes to metric spaces & how just a few results following from these are what really matters. Regardless of how true that is (I think he's biased because he is a functional analyst (analysist?)) I found that bit of intuition & motivation extremely clarifying & I'd read a lot of links, wiki's etc... just looking for that kind of motivation but it was nowhere to be found beforehand.
Similarly, I've read a lot about the implicit/inverse function theorems but I just don't understand the reasons for putting so much focus on them, but that's because I don't really know what they are saying & that's partly because I don't understand what you need to know to really appreciate these theorems.
I guess what I'm asking for is an insightful & human explanation of what these theories are, what you need to know to lead up to them, why you need to know that certain material & why these things are so powerful (for instance, I believe you can use these to prove the Lagrange Multipliers theorem, though I don't know why it has anything to do with it, I also know that knowing one means you can prove the other & that it doesn't matter which direction you approach from, but again I don't appreciate why that is).
I'm more interested in the surrounding theory, i.e. what you need to know, why you need to know that & why it is so important, than what the theorem says so I'd much rather prefer to have the motivation so that I could prove this myself.
(Yes I've read the wiki page, the threads on this site, the many articles apparently trying to motivate it, I just feel I haven't read anything that directly quells my aforementioned concerns so I think that justifies the thread).
Thanks for your time.