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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
1 year, 6 months 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. · 1 year, 6 months 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. · 1 year, 5 months ago

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That's an interesting question. Any idea what the minimum value is? Staff · 1 year, 6 months ago

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Minimum value? · 1 year, 6 months ago

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Wolfram is giving $$\approx 2.19032$$ as minimum value at $$x = -y \approx 1.0465$$. · 1 year, 5 months ago

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Comment deleted Jul 25, 2015

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Could you please elaborate? By your assumption, the minimum value will be $$r$$ but I still didn't get it. · 1 year, 6 months ago

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Oh crap, I read the question wrongly... Let me think again. · 1 year, 6 months ago

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