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Extreme Value Theorem & MVT/Rolles Theorem

  1. Mar 27, 2009 #1
    1. The problem statement, all variables and given/known data

    If f is a continous function on the closed interval [a,b], which of the following statements are NOT necessarily true?

    I. f has a minimum on [a,b].

    II. f has a maximum on [a,b].

    III. f'(c) = 0 for some number c, a < c < b

    2. Relevant equations

    Extreme Value Theorem (EVT) - the EVT states that if f(x) is continous on [a,b] there is one absolute maximum and one absolute minimum in [a,b].

    3. The attempt at a solution

    By the EVT... I believe statement "I" and statement "II" are always true.

    Statement "III" is not necessarily true. The Mean Value Theorem/Rolles Theorem states that there is a c where f'(c) = 0 iff f(x) is continous on [a,b] and iff f(x) is differentiable on (a,b). Since we weren't given the differentiability option... this is NOT necessarily true all the time.

    Is this correct? Statement III is not necessarily true?
  2. jcsd
  3. Mar 27, 2009 #2
    I, II are always true, III is not always true for example f(x) = x, [a,b] = [0,1], but you're wrong about Rolle's theorem.
  4. Mar 27, 2009 #3
    Thank you qUzz!
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