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Rolle theorem

  1. Nov 16, 2008 #1
    1. The problem statement, all variables and given/known data

    the rolle theorem said :
    (i) F(X) is continous on colse interval (a,b)

    (ii) F(X) is differentiable on open interval ( a,b)

    (iii) F(a)=F(b)

    then there is c on (a,b) such that F`(c)=0

    the question is give an example :

    1- satisfied (i) (ii) and not satisfy (i) and explain why c is not on the interval (a,b)?

    2-satisfied (i) (iii) and not satisfy (ii) and explain why c is not on the interval (a,b)?

    3-satisfied (ii) (iii) and not satisfy (i) and explain why c is not on the interval (a,b)?

    I got answers each 1 and 2 but on 3 how can I got it
    because if the function discontinous then so it is non diffrentiable on (a,b).
  2. jcsd
  3. Nov 16, 2008 #2


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    That's not a closed interval. You mean [a, b].

    But, as I pointed out above that is NOT a closed interval. Can you think of a function that is differentiable on (a, b) but NOT continuous on [a, b]? Of course, that depends precisely upon the difference between (a, b) and [a, b].
  4. Nov 16, 2008 #3
    yes I mean closed interval [a,b]

    for example F(x)= 1/x on [0,2]

    F(x) is discontinuos at x=0

    F(x) is differentiable on (0,2)

    and thank you so much
  5. Nov 17, 2008 #4
    This example does not satisfy (iii) since [itex]?=F(0)\neq F(2)=1/2[/itex]
  6. Nov 17, 2008 #5


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    Try F(x)= 1/x on (0, 2], F(0)= 1/2.
  7. Nov 18, 2008 #6
    yes it is satisfyed the theorem
  8. Nov 21, 2008 #7
    I still thinking about this question

    but we know if the function is not continuous then it is not diffrentiable

    so the answer of question (3) is will be no example because he want the function be discontinuous on interval [a,b] and diffrentiable on (a,b) that's never ever will happen in mathematics
  9. Nov 21, 2008 #8
    The function in our example is continous everywhere except at zero. At zero however, it is not required to be differentiable, so no problem.
  10. Nov 21, 2008 #9
    could you give other example :

    the example will be in discontinuoty on [a,b]
    but differentiable on (a,b)


    I will be greatfull for you if give me the example
    because this qeustion from the homework and tomorrow is the due day
  11. Nov 21, 2008 #10


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    You're correct, to the point where if the function is differentiable on (a,b) it's continuous on (a,b). Hence the only discontinuities can be at the endpoint. Examples are usually just continuous functions on (a,b) with the endpoints f(a) and f(b) redefined. Such as (a=0,b=1)

    f(x)=x on (0,1) f(0)=f(1)=1
    f(x)=sin(x) on (0,1) f(0)=2 f(1)=-3
    f(x)=ex on (0,1) f(0)=1 f(1)=0
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