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    show that:


    has only one real root.

    How would you do that without drawing it?

    (Original post by Freddy C)
    show that:


    has only one real root.

    How would you do that without drawing it?
    Let f(x) = 3x - 1 - cos2x

    Then f'(x) = 3 + 2sin2x

    Note that f'(x) >0 and so the function is always increasing

    In particular it can have at most one root [it still could have no roots]

    But note that f(0) = -2 and f(1) = 3-1-cos2 >0

    and so the function has a root between 0 and 1.
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