1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Monotonicity of convex function

  1. Apr 17, 2010 #1
    [tex]f:(a,\infty)->R[/tex]
    i want to prove, that, if function is convex, then:

    if exist [tex]x_1 \in R[/tex], exist [tex]x_2>x_1[/tex] : [tex]f(x_2)>f(x_1)[/tex]
    then:
    for all [tex]x_3>x_2[/tex] for all[tex]x_4>x_3[/tex] : [tex]f(x_4)\ge f(x_3)\ge f(x_2)[/tex]

    in other words:
    convex function is either decreasing on whole domain, or it starts to increase from some point and then is increasing from that point to the end of domain

    We dont assume that function is differential.

    With assumption that [tex]f(x_2)>f(x_1)[/tex] i should get something from definition of convex function, but i don't know how to do it.

    I'm asking for some tip.
     
    Last edited: Apr 17, 2010
  2. jcsd
  3. Apr 17, 2010 #2
    First show that convex implies this: If [itex]a < b < c[/itex], then
    [tex]
    \frac{f(b)-f(a)}{b-a} \le \frac{f(c)-f(b)}{c-b}
    [/tex]
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Monotonicity convex function Date
A Monotonic spatial function Mar 25, 2018
I Partitions of Euclidean space, cubic lattice, convex sets Mar 2, 2016
Applied maths, monotonic function Aug 17, 2015
Sequences - monotonic or not Apr 2, 2012
Monotonically increasing/decreasing functions Sep 2, 2011