Results 1 to 3 of 3

Math Help - Real Analysis

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    4

    Real Analysis

    If x>0, show that 1 + x + x^2/2 < e^x < 1 + x + x^2/2 e^x
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by dhhnerd View Post
    If x>0, show that 1 + x + x^2/2 < e^x
    I do the easier inequality, hopefully, the second one is similar.

    First we will prove that 1+x < e^x for x>0.
    Define f(x) = e^x - x - 1.
    Then f(0) = 0 but f'(x) = e^x - 1 > 0 for x>0.
    Therefore, f is increasing for x>0.
    But since we have f(0)=0 it must mean that f(x) > 0 for x>0.
    Therefore, e^x - x - 1 > 0 \implies e^x > 1+x.

    Second define g(x) = e^x - \tfrac{1}{2}x^2 - x - 1.
    Then g(0)=0 but g'(x) = e^x - x - 1>0 for x>0 by above.
    Therefore, g is increasing for x>0.
    But since we have g(0)=0 it must mean that g(x)>0 for x>0.
    Therefore, e^x > \tfrac{1}{2}x^2 + x + 1
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by dhhnerd View Post
    If x>0, show that 1 + x + x^2/2 < e^x < 1 + x + x^2/2 e^x
    What is the second inequality? e^x<1+x+\frac{x^2}{2}+e^x?

    Or e^x<1+x+\frac{x^2}{2}\cdot{e^x}

    Both are very simple
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Real analysis help please
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 19th 2009, 05:31 PM
  2. real analysis
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 21st 2009, 01:26 PM
  3. real analysis
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 19th 2009, 12:44 PM
  4. real analysis
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: August 18th 2009, 12:32 AM
  5. Real Analysis
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 12th 2008, 05:34 PM

Search Tags


/mathhelpforum @mathhelpforum