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Math Help - mean value theorem

  1. #1
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    mean value theorem

    let  f: [0,2] \to \mathbb{R} continuous on [0,2] and differentiable on (0,2) with  f(0)=f(1)= 1 \ and , f(2)= 3
    use the mean value theorem and Darboux's theorem to prove that there is  c \in (0,2 ) \ with f'(c)= \frac{1}{5}
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by flower3 View Post
    let  f: [0,2] \to \mathbb{R} continuous on [0,2] and differentiable on (0,2) with  f(0)=f(1)= 1 \ and , f(2)= 3
    use the mean value theorem and Darboux's theorem to prove that there is  c \in (0,2 ) \ with f'(c)= \frac{1}{5}
    What have you done?!?! Do you even check your answers or are you some kind of hellish poltergeist who just posts random questions?
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  3. #3
    Senior Member Shanks's Avatar
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    since f(0)=f(1), by the mean vaule theorem there exist a point s in (0,1) such that f'(s)=0,
    similarly there exist a point t in (1,2) such that f'(t)=2.
    thus by the Darboux's theorem, there exist a point c in (s,t) such that f'(c)=0.2 QED.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Shanks View Post
    since f(0)=f(1), by the mean vaule theorem there exist a point s in (0,1) such that f'(s)=0,
    similarly there exist a point t in (1,2) such that f'(t)=2.
    thus by the Darboux's theorem, there exist a point c in (s,t) such that f'(c)=0.2 QED.
    It is good practice to not answer the question after someone else asks what they have done.
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  5. #5
    Senior Member Shanks's Avatar
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    Quote Originally Posted by Drexel28 View Post
    It is good practice to not answer the question after someone else asks what they have done.
    Thanks for your advice!
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