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Math Help - Continuity

  1. #1
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    Continuity

    Prove: If f(x) is a continuous function on a closed interval [a, b] and x1, x2, ..., xn are n points in the interval, then there exists c such that:
    f(c)=\frac{f(x1)+f(x2)+...+f(xn)}{n}

    Any ideas?
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  2. #2
    Super Member girdav's Avatar
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    Re: Continuity

    Hint: intermediate value theorem, showing that the average of f(x_j) is between \min_{[a,b]} f and \max_{[a,b]}f.
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  3. #3
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    Re: Continuity

    Quote Originally Posted by loui1410 View Post
    Prove: If f(x) is a continuous function on a closed interval [a, b] and x1, x2, ..., xn are n points in the interval, then there exists c such that:
    f(c)=\frac{f(x1)+f(x2)+...+f(xn)}{n}
    Note how to use subscripts, [TEX]f(c)=\frac{f(x_1)+f(x_2)+...+f(x_n)}{n}[/TEX] gives f(c)=\frac{f(x_1)+f(x_2)+...+f(x_n)}{n}.

    Let M=\max\{f(x_n)\}\text{ and }m=\min\{f(x_n)\}. Do you see that m\le\frac{f(x_1)+f(x_2)+...+f(x_n)}{n}\le M~?.

    Can you use the intermediate value theorem?
    Thanks from loui1410
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