Can someone explain to me exactly why the Mean Value Theorem

What sort of answer are you looking for?

You seem to know applications.
It's common for existence statements with simple condition to be relevant shortcuts. Like the fixed point theorems.

>>8606672
>has a few other applications; but is that it?
I don't understand your point. What else do you want? Do you wish for it to keep your beer cold? Like seriously, what else do you expect from a theorem other than using it in proofs for other theorems or for direct application, what else could there possibly be?

>>8606672
It means if you throw something at a wall, it will hit the wall.

>>8606672
There is a point in the interval which has the same instantaneous rate of change as the average rate of change over the entire interval. Isn't that interesting in and of itself?

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114KB, 816x679px

> [math] c\in{\mathbb R} [/math]

Well...it's interesting in that it's false.

Like
https://en.wikipedia.org/wiki/Constructive_analysis#Examples

>>8607294

lol, this

it's pretty much self-evident. draw a few graphs and you'll see

>>8606672
It is the first theorem that really tells you you why derivatives are important. It relates the behaviour of a function to that of its derivative in a very straightforward way. It explains why functions with positive derivative are increasing, why functions with zero derivatives are constant and more generally allows you to get information about the variations of a function on an interval from the knowledge of its derivative and its values at the boundary.