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Math Help - Continuity and Roots tough question

  1. #1
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    Continuity and Roots tough question

    I've got a question to use the notion of continuity to show that x^7=x-2 has at least one real root, that is, there is at least one value of x that solves this equation.

    If I try to zero one side, I'm still not sure how to get the roots"
    x^7-x+2=0

    I think it involves the continuity definition which I am aware of below (trying to get the x goes to a below lim)
    \lim{x \to a}f(x)=f(a)

    So I don't know how that will fit in with my problem above. How would I find the solution to it then?
    Last edited by Kataangel; October 4th 2009 at 03:30 PM. Reason: x+2 not x-2
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    Quote Originally Posted by Kataangel View Post
    I've got a question to use the notion of continuity to show that x^7=x-2 has at least one real root, that is, there is at least one value of x that solves this equation.
    Let f(x)=x^7-x-2.
    Now f is continuous and f(1)=-3~\&~f(2)=124
    That tells us that there is root between 1~\&~2.
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  3. #3
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    Quote Originally Posted by Kataangel View Post
    I've got a question to use the notion of continuity to show that x^7=x-2 has at least one real root, that is, there is at least one value of x that solves this equation.

    If I try to zero one side, I'm still not sure how to get the roots"
    x^7-x-2=0

    I think it involves the continuity definition which I am aware of below (trying to get the x goes to a below lim)
    \lim{x \to a}f(x)=f(a)

    So I don't know how that will fit in with my problem above. How would I find the solution to it then?
    let f(x) = x^7 - x - 2

    note that f(1) = 1 - 1 - 2 = -2 < 0

    note that f(2) = 2^7 - 2 - 2 = 124 > 0

    since f(x) is continuous between x = 1 and x = 2, what function value should exist between f(1) = -2 and f(2) = 124 ?
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  4. #4
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    Sorry it seems that the actual equation is:
    x^7-x+2=0

    So now I've just calculated that f(-1)= 2 and f(-2)=-124.

    Is there an actual way I can do this more formally than just substituting 2 close numbers?
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  5. #5
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    Quote Originally Posted by Kataangel View Post
    Sorry it seems that the actual equation is:
    x^7-x+2=0

    So now I've just calculated that f(-1)= 2 and f(-2)=-124.

    Is there an actual way I can do this more formally than just substituting 2 close numbers?
    you are not required to find the actual root ... all you need do is prove that it exists.
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