Results 1 to 3 of 3

Math Help - inequation with an integral

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    34

    inequation with an integral

    Hi! Does anybody know how one can proof that \frac{c}{1+c^2}\cdot exp(-\frac{c^2}{2}) \leq \int_c^\infty exp(-\frac{z^2}{2}) dz for c>0
    I am thankful for any ideas.
    Last edited by gammafunction; May 17th 2009 at 12:02 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by gammafunction View Post

    Hi! Does anybody know how one can proof that \frac{c}{1+c^2}\cdot exp(-\frac{c^2}{2}) \leq \int_c^\infty exp(-\frac{z^2}{2}) dz for c>0
    I am thankful for any ideas.
    let z=c\sqrt{2x+1}. then I=\int_c^{\infty} \exp \left(\frac{-z^2}{2} \right) \ dz=c \exp \left(\frac{-c^2}{2} \right) \int_0^{\infty} \frac{e^{-c^2x}}{\sqrt{2x+1}} \ dx. but we know that e^a \geq 1+a, for any a \geq 0. thus \frac{1}{\sqrt{2x+1}} \geq e^{-x} and hence:

    I \geq c \exp \left(\frac{-c^2}{2} \right) \int_0^{\infty} e^{-(1+c^2)x} \ dx=\frac{c}{1+c^2} \exp \left(\frac{-c^2}{2} \right).

    it was a little tricky, wasn't it?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2008
    Posts
    34
    Tricky and wonderful . What a nice proof. Thank you a thousand times.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. An inequation related integral
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 25th 2011, 03:01 AM
  2. Help on this inequation
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: July 17th 2010, 12:22 PM
  3. Help with an inequation.
    Posted in the Algebra Forum
    Replies: 1
    Last Post: February 28th 2010, 03:21 AM
  4. Inequation
    Posted in the Algebra Forum
    Replies: 6
    Last Post: January 7th 2010, 01:40 AM
  5. inequation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 7th 2008, 09:10 PM

Search Tags


/mathhelpforum @mathhelpforum