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Mean Value theorem.

  1. Nov 23, 2012 #1
    1. The problem statement, all variables and given/known data

    1) Let f be a function differentiable two times on the open interval I and a and b two numbers in I

    Prove that: [tex]\exists c\in]a,b[:\frac{f(b)-f(a)}{b-a}=f'(a)+\frac{b+a}{c}f''(c)[/tex]

    2) Let f be a function differentiable three times on the open interval I and a and b two numbers in I.

    Prove that: [tex]\exists c\in]a,b[:f(b)=f(a)+(b-a)f'(a)+\frac{(b-a)^{2}}{2}f''(a)+\frac{b-a}{2}f'''(c)[/tex]




    3. The attempt at a solution

    Any tips on how to start please. Thank you in advance.
     
  2. jcsd
  3. Nov 23, 2012 #2

    micromass

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    What did you try?

    Are you reminded of some general result or theorem?
     
  4. Nov 23, 2012 #3
    Well the theorem states that if a function is continuous on a closed interval [a,b] and is differentiable on the open interval (a,b) then there exists a c in the open interval (a,b) such that.

    [tex]f'(c)=\frac{f(b)-f(a)}{b-a}[/tex]

    And then i tried counting the second derivative but i couldn't get anything out of it.
     
  5. Nov 23, 2012 #4

    micromass

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    Do you know Taylor's theorem?
     
  6. Nov 23, 2012 #5
    Yes i know that it can be solved using taylor's theorem easily, but we need to prove it for the a function differentiable twice and a function differentiable 3 times and then we have to prove taylor's theorem for a function differentiable n times. So I thought that since there exists a C in the open interval (a,b) then f(c) will have to be the mean of the f(b)+f(a) and when i take the derivative of that i get f'(c)=(f'(b)-f'(a))/2, but I don't know what to do from here on.
     
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