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How to proove that that e^x is convex

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data

    I have to determine if [e][/x] is a convex function. If it is then show proof. I know its a convex function by looking at the graph, Iam stuck at prooving it mathematically though.


    2. Relevant equations

    The function is f(x)=e^x.


    3. The attempt at a solution
    I am certain that the function is convex. I'am having trouble proving it though.

    Assuming we pick a point and call it x0, then a lies to the left of x0 and point b lies to the right.

    f(ta+(1-t)b)<=t*f(a)+(1-t)*f(b)

    Once we substitute we get.
    e^((ta+(1-t)b))<=t*e^a+(1-t)*e^b

    I'am stuck at proving how this inequality is true.

    Thanks!
     
  2. jcsd
  3. Oct 2, 2011 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If you are required to use the definition of "convex" then use the fact that
    [tex]e^{ta+ (1-t)b}= e^{ta}e^be^{-bt}[/tex]

    Or are you allowed to use the fact that a function is convex if and only if its second derivative is positive for all x?
     
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