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Inequality?

Let \(x\) and \(y\) be real number with \(xy \neq -1,\) and \[\displaystyle \frac{x^7y+xy^7}{1+x^5y^5} = 5\] What is the minimum value of \[\displaystyle x^2+y^2\]

I've been using Lagrange Multiplier but can't get the answer, is there any inequality theorem I could use?

Note by Natasha Andriani
2 years ago

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Those equation can be expressed as \[\frac{xy(x^2+y^2)((x^2+y^2)^2-3x^2y^2)}{1+x^3y^3}=5\] Stuck here. I'll add if I have something. Fariz Azmi Pratama · 2 years ago

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Friend try this,

Let \(xy = t\) \[\frac { t({ x }^{ 6 }+{ y }^{ 6 }) }{ { t }^{ 5 }+1 } =5\\ t({ x }^{ 6 }+{ y }^{ 6 })=5t^{ 5 }+5\\ { x }^{ 6 }+{ y }^{ 6 }=5{ t }^{ 4 }+\frac { 5 }{ t } \]

Now using Newtonian Sum you may find the value of \(x^{6}+y^{6}\) in terms of \(x\) and \(y\) you may find the answer then. Lakshya Sinha · 1 year, 11 months ago

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That's an interesting question. Any idea what the minimum value is? Calvin Lin Staff · 2 years ago

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@Natasha Andriani Wolfram is giving \(\approx 2.19032\) as minimum value at \(x = -y \approx 1.0465\). Krishna Sharma · 1 year, 11 months ago

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Comment deleted Jul 25, 2015

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@Pi Han Goh Could you please elaborate? By your assumption, the minimum value will be \(r\) but I still didn't get it. Natasha Andriani · 2 years ago

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@Natasha Andriani Oh crap, I read the question wrongly... Let me think again. Pi Han Goh · 2 years ago

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