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Math Help - cauchy theorim question..

  1. #1
    MHF Contributor
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    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??
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  2. #2
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    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].
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