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Math Help - Prove A Function Is Convex?

  1. #1
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    Prove A Function Is Convex?

    Prove the function:

    -sum_i(x_i*ln(x_i))

    is convex for x>0

    I have graphed the function, and can see visually that it is convex, but I don't know how to prove that it is convex. If I graph the slope of the graph between any two points, it seems like the slope is always decreasing. This seems like a way to prove it visually, but how can I put that into a mathematical proof?
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  2. #2
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    Quote Originally Posted by jjpioli View Post
    Prove the function:

    -sum_i(x_i*ln(x_i))

    is convex for x>0

    *** "x" ? Which x? You're apparently defining a function of several variables or what...??

    Tonio



    I have graphed the function, and can see visually that it is convex, but I don't know how to prove that it is convex. If I graph the slope of the graph between any two points, it seems like the slope is always decreasing. This seems like a way to prove it visually, but how can I put that into a mathematical proof?
    .
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  3. #3
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    tonio, sure, I hope I can better explain this:

    Here is a link to the function: sum of x*ln(x) - Wolfram|Alpha (my tex skills are bad)
    So basically, I understand that it is convex from visual inspection, but I don't know how to prove it is mathematically.
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  4. #4
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    Quote Originally Posted by jjpioli View Post
    tonio, sure, I hope I can better explain this:
    Here is a link to the function: sum of x*ln(x) - Wolfram|Alpha (my tex skills are bad)
    So basically, I understand that it is convex from visual inspection, but I don't know how to prove it is mathematically.
    In some real sense, a infinite series is a discrete graph.
    That is, it is a countable collection of points (n,S_n) where S_n is the nth partial sum of the series.
    You have used the term function and asked about it being convex.
    I don't follow you question here.
    Clearly the sequence S_N=\sum\limits_{n = 1}^N {n\ln (n)} is an increasing sequence.
    Its discrete graph is convex by definition.
    What am I missing?
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