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Math Help - Rolle's Theorem

  1. #1
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    Rolle's Theorem

    Show that 1+2x+x^3+4x^5=0 has exactly one root using Rolle's Theorem.
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  2. #2
    MHF Contributor Calculus26's Avatar
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    1)Consider f ' (x) = 2 + 3x^2 + 20x^4

    Using the quadratic formula there are no zeroes of the derivative

    2) If f(x) had more than one zero say x =a and b then by Rolles Thm there would be a number c in(a,b) st f ' (c) = 0 which contradicts 1)

    3) Since f(x) is a fifth degree polynomial it has at least one zero

    Therefore combining 2) and 3) f has exactly one zero
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