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    Mean Value Theorem : Roots of Derivatives on an Interval

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    The function f (x) and all of its derivatives are continuous on [0, 10]. You know that f (0) = 0,
    f (2) = 0, f (3) = 0, f (6) = 0, and f (8) = 0. At how many points must the first derivative of f (x)
    be zero? At how many points must the second derivative of f (x) be zero? At how many points must the third derivative of f (x) be zero? And so on. Justify your answers.

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    Solution Preview

    We can use the mean value theorem.
    f(0)=0, f(2)=0, f(3)=0, f(6)=0, f(8)=0
    1. For the first derivative, we can find x1 in (0,2), such that f'(x1)=0; we can find x2 in (2,3), such that f'(x2)=0;
    we can find x3 in (3,6), such that f'(x3)=0; we can find ...

    Solution Summary

    Mean value theorem and roots of derivatives on an interval are investigated. The solution is detailed and well presented.