# Prove A Function Is Convex?

• May 1st 2011, 06:47 AM
jjpioli
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?
• May 1st 2011, 09:29 AM
tonio
Quote:

Originally Posted by jjpioli
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?

.
• May 1st 2011, 01:27 PM
jjpioli
tonio, sure, I hope I can better explain this:

Here is a link to the function: sum of x&#42;ln&#40;x&#41; - 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.
• May 1st 2011, 01:48 PM
Plato
Quote:

Originally Posted by jjpioli
tonio, sure, I hope I can better explain this:
Here is a link to the function: sum of x&#42;ln&#40;x&#41; - 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 $\displaystyle (n,S_n)$ where $\displaystyle S_n$ 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 $\displaystyle S_N=\sum\limits_{n = 1}^N {n\ln (n)}$ is an increasing sequence.
Its discrete graph is convex by definition.
What am I missing?