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The mean Value Theorem

  1. Mar 14, 2017 #1
    1. The problem statement, all variables and given/known data
    Find all the numbers [itex]c[/itex] that satisfy the conclusion of the Mean Value Theorem for the functions

    [itex]f(x)=\dfrac{1}{x-2}[/itex] on the interval [itex][1, 4][/itex]

    [itex]f(x)=\dfrac{1}{x-2}[/itex] on the interval [itex][3, 6][/itex]

    I don't need help solving for [itex]c[/itex], I just want to know how I can verify that the hypotheses of the mean Value Theorem are satisfied by the function [itex]f(x)[/itex] on the given interval. I know the first one, there is no such number [itex]c[/itex] that is guaranteed by the mean Value Theorem but there is for the second one. How can we verify the first one?

    2. Relevant equations

    Rolle's Theorem: Let [itex]f[/itex] be a function that satisfies the following three hypotheses:

    1. [itex]f[/itex] is continuous on the closed interval [itex][a, b][/itex]
    2. [itex]f[/itex] is differentiable on the open interval [itex](a, b)[/itex]
    3. [itex]f(a)=f(b)[/itex]

    Then there exists a number [itex]c[/itex] between [itex]a[/itex] and [itex]b[/itex] such that [itex]f'(c)=0[/itex]

    3. The attempt at a solution

    So I would test if it's differentiable for the first function.

    [itex]f(1)=\dfrac{1}{1-2}=-1[/itex]

    [itex]f(4)=\dfrac{1}{4-2}=1/2[/itex]

    So for the first interval of [itex][1, 4][/itex] there is no such [itex]c[/itex] number because the inputs (-1 and 1/2) are not equal.
    How do we know if it's continuous? If it's differentiable does that mean it has to be continuous?
     
  2. jcsd
  3. Mar 14, 2017 #2

    mfb

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    Staff: Mentor

    The mean value theorem is more general than Rolle's theorem, it does not need the f(a)=f(b) condition.
    Yes - but keep in mind that this just helps directly with (a,b), not with [a,b].
     
  4. Mar 14, 2017 #3
    So my interval is a closed interval for [a, b]. If the case doesn't apply, how would you go about verifying if it's continuous?
     
  5. Mar 14, 2017 #4

    mfb

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    Staff: Mentor

    At the time you get those questions, it should be fine to do that just by inspection.

    But there is also a statement about the continuity of f(x)=c/g(x) with a constant c that depends on properties of g(x).
     
  6. Mar 14, 2017 #5
    Alright, thank you!
     
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