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Rolle's theorem (maybe)

  1. Oct 16, 2011 #1
    1. The problem statement, all variables and given/known data
    A function is continuous/differentiable on the interval [0,3]. I have been given that f(0)=1, f(1)=2, f(3)=2.
    I need to prove that there exists a c within the interval [0,3] such that f(c) = c.

    2. Relevant equations

    3. The attempt at a solution
    f(1)=f(3)=2. From Rolle's theorem, there exists a c within [1,3] such that f`(c)=0. From this I am able to show that at some point c within [1,3] the function must be constant i.e f(c)=k.
    I am not sure how to show that the constant is equal to the point i.e k=c.
     
  2. jcsd
  3. Oct 16, 2011 #2

    HallsofIvy

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    The problem does not ask you about the derivative of f so Rolle's theorem is unnecessary. Also it makes no sense to say that a function is constant at a point! Every function has a single value at every point of its domain.

    Define g(x)= f(x)- x. g(0)= 1- 0= 0, g(1)= 2- 1= 1, and g(3)= 2- 3= -1. That is, g(1)> 0 and g(3)< 0. What does that tell you?


    (Much of the information given in this problem is unnecessary! The function does not have to be differentiable, just continuous. The information about f(0)= 1 is irrelevant. And, in fact, the desired point must be in the interval [1, 3].)
     
  4. Oct 16, 2011 #3
    Thanks for the response. This helped a lot.
     
    Last edited: Oct 16, 2011
  5. Oct 16, 2011 #4
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