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)

#Inequalities

Easy Math Editor

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Comments

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.

Cody Johnson - 7 years, 6 months ago

See the footnote on page 7 of this document.

Cody Johnson - 7 years, 6 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.

Ahaan Rungta - 7 years, 6 months ago

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

Calvin Lin Staff - 7 years, 6 months ago

Please.

Cody Johnson - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$