These statement are often quoted as fact. But are they true?

Statement 1. $f(x,y)$ is a function that is cyclic in the variables, does this mean that subject to $ x+y = 1 $, the local minimum or maximum can only occur at $ f( \frac{1}{2}, \frac{1}{2} ) ? $

Statement 2. $f(x,y,z)$ is a function that is cyclic in the variables, does this mean that subject to $x + y + z = 1$ , the local minimum or maximum can only occur at $f( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} )$ ?

#Inequalities #Minimum

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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Comments

For counterexamples, take $f(x,y)=xy$ and $f(x,y,z)=xyz$

Daniel Chiu - 7 years, 5 months ago

That's a quick reply.

I've updated it slightly, to get at the original intention.

Chung Kevin - 7 years, 5 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}`	$\boxed{123}$