Why does the intermediate value theorem give min < x < max for inequalities?

[transgalactic]

i got this question
http://img412.imageshack.us/img412/3713/88436110xw9.gif

here is the solution:
http://img297.imageshack.us/img297/6717/14191543qm1.th.gif
they are taking the minimal value
and the maximal value
the innequalitty that the write is correct min< <max

but why??

 

[CompuChip]

Well, if you have
$$f(x_1) + f(x_2) + \cdots + f(x_n)$$
and you know that each of the $f(x_i)$ is not greater than M, then you can write
$$f(x_1) + f(x_2) + \cdots + f(x_n) \le M + M + \cdots + M = n \cdot M;$$
similarly for the minimum.

It's simply applying the inequality that
x + y <= M + y
if x <= M.

 

[transgalactic]

i agree with you
but why they do that
how is it linked to cauchy theorem
?

 

[CompuChip]

I don't know what it has to do with Cauchy's theorem, but it does have to do with the intermediate value theorem: for any value c between m and M (assuming some conditions on f which you didn't state) there is an x such that f(x) = c.