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!

Homework Help: Convexity of functions

  1. Dec 13, 2006 #1
    1. The problem statement, all variables and given/known data
    Let f be differentiable on (a,b). Show that f is convex if and only if for every x,y in (a,b), f(y)-f(x)>= (y-x)f'(x)

    3. The attempt at a solution
    The mean value theorem says that there exists an x' in (a,b) such that f'(x') is the average rate of change of the functions. So I have the equation for that tangent line. I am stuck there.
  2. jcsd
  3. Dec 13, 2006 #2


    User Avatar
    Homework Helper

    There's two directions to prove, so which one are you asking about? And what is the definition of a convex function?
  4. Dec 13, 2006 #3
    If I prove one direction, is the proof in the other direction just the logic going the other way? In any case, let's say I want to show it is convex given for every x,y in (a,b), f(y)-f(x)>= (y-x)f'(x)
  5. Dec 13, 2006 #4


    User Avatar
    Science Advisor

    I'm not sure what you mean here. You want to prove that the straight line between (a,f(a)) and (b,f(b)), which is y= (f(b)-f(a))/(b-a) (x- a)+ f(a) lies above the curve y= f(x). That is, that (f(b)-f(a))/(b-a) (x- a)+ f(a)> f(x) for all x between a and b.

    Good heaven's no! There are plenty of theorems that are true in one direction but false in the other!
  6. Dec 13, 2006 #5


    User Avatar
    Homework Helper

    That's certainly not true in general. If we stick to the direction you mentioned, you can rearrange and get:

    [tex]f(x) \geq f(x_0) + f'(x_0)(x-x_0)[/tex]

    Or in other words, f lies above every line tangent to f. From here it's easy to show the function is convex, it's just a matter of plugging in to the defintion (which I'm not going to copy for you). The other direction will be a little harder.
  7. Dec 13, 2006 #6
    Halls of Ivy,

    I meant for theorems which state if and only if. Are there if and only if statements where the logic cannot backwards?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook