# Inflection Point Inequality Theorem

So I was told by Patrick Hompe that there's some magical theorem used to prove inequalities in which you test if a function has only one inflection point, and then you prove the inequality for just one case. What's the exact theorem statement, and does it have a name? (I'm not talking about Jensen's Inequality)

Note by Cody Johnson
4 years, 7 months ago

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For all those curious, I got this as a result:

If you have some variables satisfying $$x_1+x_2+\dots+x_n=C$$ (some constant), then if $$f$$ has 1 inflection point, then $$f(x_1)+f(x_2)+\dots+f(x_n)$$ achieves all extrema when at least $$n-1$$ of the variables are equal.

I also hear that it's called the (n-1)-equal-value theorem, as shown here.

- 4 years, 7 months ago

See the footnote on page 7 of this document.

- 4 years, 7 months ago

I have used this technique in the past without a name and just written the whole thing out. The graders not only had nothing to complain about but they were also very happy.

- 4 years, 7 months ago

Can you state this technique / theorem, for those who do not know it?

Staff - 4 years, 7 months ago