# Math Help - cauchy theorim question..

1. ## cauchy theorim question..

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

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??

2. I am not exactly sure what you mean by Cauchy law.
I think what you are using here is the intermediate value theorem.
Since, $\frac{f(x_1)+...+f(x_n)}{n} \leq \frac{M+...+M}{n} = M$ and $\frac{f(x_1)+...+f(x_n)}{n} \geq \frac{m+...+m}{n} = m$
It means by the intermediate value theorem there is $a\leq c\leq b$ so that $f(c) = \frac{f(x_1)+...+f(x_n)}{n}$

Thus, you need the condition $f$ is continous on $[a,b]$.