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How do I show that a derivative of a polynomial has a zero in an interval?

  1. Jun 20, 2011 #1
    1. The problem statement, all variables and given/known data

    If an even degree polynomial of order 2n intersects the x-axis twice, how do I show that the (2n-1) th derivative has a zero in that interval?


    2. Relevant equations

    example: let g(x)=x^3(1-x). Show without computation that g'''(c) =0 for some c in (0, 1).

    3. The attempt at a solution

    I know that g' has a zero at x=1/2 because that is a maximum. I don't know how I can be sure that the 2nd and 3rd derivative have a zero in this interval.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 21, 2011 #2

    lanedance

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    Homework Helper

    consider what happens to terms less of order (2n-2) and less...
     
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