Results 1 to 4 of 4

Thread: Rolle's Theorem

  1. #1
    Junior Member
    Join Date
    Mar 2008
    Posts
    87

    Rolle's Theorem

    Hi, I am stuck to a certain point on this problem:

    1. Determine whether Rolle's Theorem can be applied to f on the closed interval [a,b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a,b) such that f'(c)=0.

    f(x)=(x-1) (x-2) (x-3), [1,3]

    I know Rolle's Theorem can be applied since f is continuous along the interval of [1,3] and differentiable along that interval as well.

    f(x) = x[sup:cghctzop]3[/sup:cghctzop]-6x[sup:cghctzop]2[/sup:cghctzop]+11x-6
    f'(x) = 3x[sup:cghctzop]2[/sup:cghctzop]-12x+11

    I'm stuck after that, how would you find the values of c? Thanks!

  2. #2
    Elite Member
    Join Date
    Jun 2007
    Posts
    12,820

    Re: Rolle's Theorem

    Quote Originally Posted by Violagirl
    Hi, I am stuck to a certain point on this problem:

    1. Determine whether Rolle's Theorem can be applied to f on the closed interval [a,b]. If Rolle's Theorem can be applied, find all values of c in the open interval (a,b) such that f'(c)=0.

    f(x)=(x-1) (x-2) (x-3), [1,3]

    I know Rolle's Theorem can be applied since f is continuous along the interval of [1,3] and differentiable along that interval as well.

    f(x) = x[sup:bmcxsu99]3[/sup:bmcxsu99]-6x[sup:bmcxsu99]2[/sup:bmcxsu99]+11x-6
    f'(x) = 3x[sup:bmcxsu99]2[/sup:bmcxsu99]-12x+11

    I'm stuck after that, how would you find the values of c? Thanks!
    find all values of c in the open interval (a,b) such that f'(c)=0.
    Read your problem statement carefully.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
    Senior Member
    Join Date
    Mar 2009
    Posts
    1,577

    Re: Rolle's Theorem

    [tex]Rolle's \ Theorem[/tex]

    [tex]Let \ f \ be \ differential \ on \ (a,b) \ and \ continuous \ on \ [a,b].[/tex]

    [tex]If \ f(a) \ = \ f(b) \ = \ 0, \ then \ there \ is \ at \ least \ one \ point \ c \ in \ (a,b) \ where \ f \ ' \ (c) \ = \ 0.[/tex]

    [tex]f(x) \ = \ (x-1)(x-2)(x-3), \ f(1) \ = \ f(3) \ = \ 0, \ hence \ f \ ' \ (c) \ = \ 0.[/tex]

    [tex]f \ ' \ (x) \ = \ 3x^{2}-12x+11, \ f \ ' \ (c) \ = \ 3c^{2}-12c+11 \ = \ 0, \ solve \ for \ c \ and \ you \ are \ done.[/tex]

    [tex]See \ graph.[/tex]

    [attachment=0:ioe5airh]pac.jpg[/attachment:ioe5airh]
    Attached Images Attached Images
    I am not, therefore I do not think. Contrapositive of Descartes' quip.

  4. #4
    Junior Member
    Join Date
    Mar 2008
    Posts
    87

    Re: Rolle's Theorem

    Thank you both very much, I figured it out.

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •