Results 1 to 2 of 2

Math Help - generalized Rolle's theorem trouble

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    104

    generalized Rolle's theorem trouble

    Hi. I am having trouble proving the following:
    Let f be continuous on [a,b], n time differentiable on (a,b), and f(x_0)=f(x_1)=...=f(x_n)=0 for some x_0,x_1,...,x_n in [a,b]. Show that there exists c in (a,b) such that f^{(n)}(c)=0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Notice that the case n=1 is simply Rolle's theorem. Now applying Rolle to each interval of the form [x_i,x_{i+1}] with i=0,...,n we have that there exist n points, say y_0,...,y_{n-1} where the first derivative is 0, and applying Rolle again in [y_i,y_{i+1}] we get n-1 points where f^{(2)} is 0. Arguing inductively we get that there exists n-(j-1) points where f^{(j)} is zero so we get 2 points where f^{(n-1)} is zero and applying Rolle one more time the result follows.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Rolle's Theorem
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 19th 2010, 07:31 PM
  2. rolle's theorem help
    Posted in the Calculus Forum
    Replies: 6
    Last Post: November 22nd 2009, 11:07 AM
  3. intermediate value theorem/rolle's theorem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 8th 2007, 01:55 PM
  4. Rolle's Theorem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 26th 2007, 07:46 AM
  5. Generalized Mean Value Theorem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 28th 2006, 07:51 AM

Search Tags


/mathhelpforum @mathhelpforum