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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 \( f( \frac{1}{3}, \frac{1}{3}, \frac{1}{3} ) \)?

Note by Chung Kevin
3 years, 9 months ago

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For counterexamples, take \[f(x,y)=xy\] and \[f(x,y,z)=xyz\]

Daniel Chiu - 3 years, 9 months ago

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That's a quick reply.

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

Chung Kevin - 3 years, 9 months ago

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