Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Contradiction of statement regarding monotonicity

  1. Feb 12, 2012 #1
    Hi all!
    We were given to proove or falsify the following statement:

    Given [tex]f(x)>0 \,\ ,\,x>0 \,\,\,\,,\lim_{x\to\infty}f(x)=0[/tex]
    Then f(x) is strictly decreasing at certain aεℝ for every x>a

    Now in their solution they contradicted the statement with:
    #1 & \mbox{if } #2 \\
    #3 & \mbox{if } #4
    \right. } f(x) = \twopartdef { \frac{1}{2x} } {x \,\,\, rational} {\frac{1}{x}} {x \,\,\, irrational}[/tex]

    Now i thought of another one: [tex] f(x)=\frac{sin(x)+2}{x^2} [/tex]
    Is that a good example? Thank you!
  2. jcsd
  3. Feb 12, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Yes, that's a nice example too. If it is something to hand in you would want to include an argument to show that it isn't strictly decreasing for x large enough.
  4. Feb 12, 2012 #3
    Will do. Thank you!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Contradiction of statement regarding monotonicity
  1. Proof by Contradiction (Replies: 7)

  2. Monotonic polynomial (Replies: 15)

  3. Non monotonic (Replies: 2)