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Thread: Proof required.

  1. #1
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    Proof required.

    Let f(x) be a function with n distinct zeros on [a, b]. Prove that, f'(x) has at least n − 1 distinct zeros on [a, b].

    Don't really know where to start here, any help is appreciated.
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  2. #2
    Super Member PaulRS's Avatar
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    But does it say if the function is differentiable in $\displaystyle (a,b)$?

    If so, remember Rolle's Theorem, given real numbers $\displaystyle c<d$ if $\displaystyle f(x)$ is continuous in $\displaystyle [c,d]$,differentiable in $\displaystyle (c,d)$, and $\displaystyle f(c)=f(d)$ then there is a certain $\displaystyle c$ in $\displaystyle (c,d)$ such that $\displaystyle f'(c)=0$
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  3. #3
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    yes the function is differentiable on {a b} and continuous. We need to prove that there is n-1 zeros and not that there is a zero(rolle's) --> f'(c)= 0. For example the function itself may have 5 zeros on [a b] so its derivative will have 4 on [a b], that what is required to prove
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  4. #4
    Super Member PaulRS's Avatar
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    These are our zeros

    Let $\displaystyle x_1,x_2,x_3,...,x_n\in[a,b]$ such that $\displaystyle x_1<x_2<x_3<...<x_n$, and $\displaystyle f(x_i)=0$ $\displaystyle \forall{i\in{N}}$ $\displaystyle 1\leq{i}\leq{n}$

    Consider $\displaystyle [x_1,x_2]$, $\displaystyle [x_2,x_3]$ ...$\displaystyle [x_{n-1},x_n]$

    And apply Rolle's Theorem to them

    For example
    Since f(x) is continuos in $\displaystyle [x_1,x_2]$, differentiable in $\displaystyle (x_1,x_2)$ and $\displaystyle f(x_1)=f(x_2)=0$ we can assure there's a certain $\displaystyle c_1\in{(x_1,x_2)}$ such that $\displaystyle f'(c_1)=0$
    Last edited by PaulRS; Dec 5th 2007 at 08:20 AM.
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  5. #5
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    Hello, somestudent2!

    I must assume that the function is continuous on $\displaystyle [a,b].$


    Let $\displaystyle f(x)$ be a function with $\displaystyle n$ distinct zeros on $\displaystyle [a, b].$

    Prove that $\displaystyle f'(x)$ has at least $\displaystyle n - 1$ distinct zeros on $\displaystyle [a, b].$

    Maybe a sketch will help you visualize the problem . . .


    Suppose $\displaystyle f(x)$ has four distinct zeros on $\displaystyle [a,b].$
    Code:
            |
            |           --*--
            |          *     *              --*--
            |        *         *           *     *
        - - + - + - o - - - - - o - - - - o - - - -o- - - + - -
            |   a                  *     *            *   b
            |      *                 --*--
            |

    Then there are at least three horizontal tangents on $\displaystyle [a,b].$

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  6. #6
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    thank you very much guys
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