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Mean Value Theorem

  1. Nov 3, 2013 #1

    Qube

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    1. The problem statement, all variables and given/known data

    http://i.minus.com/jX32eXvLm6FGu.png [Broken]

    2. Relevant equations

    The MVT applies if

    1) The function is continuous on the closed interval [a,b] such that a<b.
    2) The function is differentiable on the open interval (a,b)

    And if the above two conditions are fulfilled then there is some point c between a and b at which the slope is equal to (f(b) - f(a)) / (b-a)

    3. The attempt at a solution

    1) The function is continuous for all real x. The function has a slope for all real x.
    2) The function is differentiable for all x, as stated in the problem.

    Therefore the MVT applies.

    Because the MVT applies [f(7) - f(1)] / 6 = f'(c).

    The maximum that f'(c) can be is 5, as stated in the problem. The slope is always between 2 and 5, including the endpoints. The minimum f'(c) can be is 2.

    Therefore the inequality should be 12 ≤ f(7) - f(1) ≤ 30.
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Nov 3, 2013 #2

    LCKurtz

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    I don't see any question. But I do like that last inequality, if you were wondering.
     
    Last edited by a moderator: May 6, 2017
  4. Nov 4, 2013 #3

    Qube

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    Alright. That was what I was looking for. Thank you :)!
     
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