# Cauchy theorim question

1. Jan 20, 2009

### transgalactic

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

here is the solution:

they are taking the minimal value
and the maximal value
the innequalitty that the write is correct min< <max

but why??

Last edited by a moderator: Apr 24, 2017 at 10:28 AM
2. Jan 20, 2009

### 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.

3. Jan 20, 2009

### transgalactic

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

4. Jan 20, 2009

### 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.