Results 1 to 4 of 4

Math Help - intermediate value theorem

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    10

    intermediate value theorem

    Let f (x) be a monic polynomial of odd degree. Prove that for some A < 0,
    f (A) < −1 and for some B > 0, f (B) > 1. Deduce that every polynomial of odd degree has a real root.

    Suppose f(x) =x^(2n+1)+a2n x^(2n)...+a1x+a0, but I've no clue how to go from here. Could anyone please give me some hints? Any help is appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2009
    Posts
    130
    Factor out  x^{2n + 1} and then look at values of the resulting expression for very large x.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    10
    f(x)=(x-1)(x^2n+x^(2n-1)...+1)+1+...>1 when x is sufficiently large
    f(x)=(x+1)(x^2n-x^(2n-1)+...+1)-1+...<-1 when x is sufficiently small
    Is that what you are talking about? I'm still a bit unsure
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Aug 2009
    Posts
    130
     f(x) = x^{2n + 1} + a_{2n} x^{2n} + \cdots + a_1x + a_0 = x^{2n+1} (1 + a_{2n} \frac{1}{x} + a_{2n - 1} \frac{1}{x^2} + \cdots + a_1 \frac{1}{x^{2n}} + a_0 \frac{1}{x^{2n+1}}) .

    For very large positive x, the expression in the brackets is very close to 1, right? And the  x^{2n+1} is a very large positive value.

    For very large negative x, the expression in the brackets again is very close to 1 and the  x^{2n+1} is a very large negative value, since the exponent is negative.

    So you know that for large enough positive x, f(x) > 0 and for large enough negative x, f(x) < 0. Now apply the intermediate value theorem.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Intermediate Value Theorem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 5th 2010, 08:00 PM
  2. Intermediate Value Theorem
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: November 7th 2008, 10:14 AM
  3. Replies: 6
    Last Post: October 28th 2008, 04:18 PM
  4. Intermediate Value Theorem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 8th 2008, 07:48 PM
  5. intermediate value theorem/rolle's theorem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 8th 2007, 01:55 PM

Search Tags


/mathhelpforum @mathhelpforum