Results 1 to 4 of 4
Like Tree1Thanks
  • 1 Post By johng

Math Help - Use Rolle's theorem to prove function cannot have 5 zeros.

  1. #1
    Junior Member
    Joined
    Jan 2013
    From
    australia
    Posts
    39

    Question Use Rolle's theorem to prove function cannot have 5 zeros.

    I have to show that Rolle's theorem can be used to prove that the function 2x5 − 5x4 + ax cannot have five distinct real zeros. I know that Rolle's theorem means that if f(a)=f(b)=0 there must be some x value c between a and b such that f'(c)=0. I am unsure of how to do this, would i show that the derivative does not equal zero 4 times and then use Rolle's theorem to say therefore there must be less than 5 zeros for the function.


    Would this be a way to do it? If not how would i do it?


    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1

    Re: Use Rolle's theorem to prove function cannot have 5 zeros.

    Quote Originally Posted by mcleja View Post
    I have to show that Rolle's theorem can be used to prove that the function 2x5 − 5x4 + ax cannot have five distinct real zeros. I know that Rolle's theorem means that if f(a)=f(b)=0 there must be some x value c between a and b such that f'(c)=0. I am unsure of how to do this, would i show that the derivative does not equal zero 4 times and then use Rolle's theorem to say therefore there must be less than 5 zeros for the function.
    Look at the following>
    [tex]2x^5-4x^4+ax[/tex] gives 2x^5-4x^4+ax

    Isn't that a lot easier to read than "2x5 − 5x4 + ax". It takes so little effort to learn to do the LaTeX code.
    Click on the “go advanced” tab. On the toolbar you will see \boxed{\Sigma} clicking on that give the LaTeX wraps, [tex] [/tex]. The code goes between them.

    You will find that helpers are more likely to reply to well formatted questions.

    Now did i read your question correctly?
    Last edited by Plato; February 9th 2013 at 08:38 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    611
    Thanks
    248

    Re: Use Rolle's theorem to prove function cannot have 5 zeros.

    If g is any differentiable function with zeros x_1<x_2<x_3<x_4<x_5, Rolle's Theorem guarantees the derivative g^\prime has 4 zeros. (Don't you see why?) Similarly, if g has n zeros, g^\prime has n-1 zeros.

    Now let f(x)=2x^5-5x^4+ax. If f had 5 real zeros, then f^\prime has 4 zeros and so f^{\prime\prime} would have 3 zeros. But f^{\prime\prime}(x)=40x^3-60x^2=x^2(40x-60) clearly has only 2 zeros. So f can not have 5 zeros.

    I think using Latex with this editor (or any editor which requires HTML tags) is a real pain. Notice I was "lazy" and didn't put tags around some of the numbers, n and n-1, so the fonts are different. I do agree for short expressions (a polynomial for example), it's pretty easy. So I'd say use enough Latex so that the text is readable, but don't worry about trying to produce "professional" copy.
    Thanks from emakarov
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Jan 2013
    From
    australia
    Posts
    39

    Re: Use Rolle's theorem to prove function cannot have 5 zeros.

    Thanks Plato and johng, i will try to use latex from now on. And thanks to you johng the problem seems crystal clear to me now, thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using Rolle's theorem to prove at most one solution of quadratic
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: February 11th 2011, 06:22 AM
  2. Prove using (ONLY) Rolle's Theorem (??)
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: January 4th 2010, 11:04 AM
  3. Replies: 4
    Last Post: November 3rd 2009, 11:33 AM
  4. intermediate value theorem/rolle's theorem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 8th 2007, 01:55 PM
  5. Rolle's Theorem Help :D
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 18th 2007, 09:38 AM

Search Tags


/mathhelpforum @mathhelpforum