Q:

What are the non-obvious corollaries that are proved with the mean value theorems?

I'm reading Stromberg Introduction to Classical Real Analysis, in the section of the mean-value theorems. He argues the following:

I am reading the section and doing the exercises, it will take some time to answer all of them but I feel deeply curious about these non-obvious corollaries - I've briefly looked in the book but I still didn't find them (perhaps they are left as exercises), I also seen the same affirmation about it's importance on other books but didn't read enough of the books to find the answer and some of these theorems were presented to me in my calculus classes, but again, there was no explanation of these further corollaries. So could you briefly point me to them?

A:

Fundamental theorem of calculus allows you to determine the function if you know its derivative. 
Mean value theorem allows you to estimate the function if you can estimate its derivative. 

Drawing this parallel, the inequality 
$$|f(a)-f(b)|\le \sup_{[a,b]}|f'|\, |a-b| \tag1$$ 
could be called the fundamental inequality of calculus.
Example of usage: for some reason you want to know if the solution of differential equation $y'=f(y)$ is unique and well-behaved. Maybe $y(t)$ is the number of slow neutrons in your nuclear reactor at time $t$... There's a theorem that says this is so if $f$ satisfies the Lipschitz condition $|f(a)-f(b)|\le L|a-b|$. If you know (1), then you calculate $f'$, show it's bounded, and go home happy. Otherwise you are stuck at work at night, trying to prove some ugly two-variable inequalities while your reactor overheats.   :) 

