advanced english speeches essays

NIKE 57 · 29.02.2012

$Consider: \\$f(x,y,z):=\frac{1}{1+x^3}+\frac{1}{1+y^3}+\frac {1}{1+z^3}-\frac{3}{1+xyz}$\\ for $x,y,z>1.$ \\ \\ So $\frac{\partial f}{\partial x}=3(g(yz)-g(x^2))$\\ \\ where \\ \\ $g(t):=\frac{t}{(1+xt)^2}.$\\ \\ Calculus shows that $g$ is decreasing for $t>1$, no matter what $x$ is.\\ \\ Consequently, if we keep $y,z$ fixed and vary $x$, we have that $f$ is decreasing for $x\leq \sqrt{yz}$ and increasing for $x\geq \sqrt{yz}$. Similar results hold for $y,z$ by symmetry.\\ \\ By symmetry, we can assume $x\geq y \geq z.$ Hence, noting that $\sqrt{xy}\geq y \geq z$ and applying the above monotonicity result in $z$ and then $x$ yields:\\ \\ $f(x,y,z)\geq f(x,y,y)\geq f(y,y,y) = 0.$\\ \\ This completes the proof.$

advanced english speeches essays

Advanced english speeches essays

ALMAZIK67 · 12.02.2014

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Advanced english speeches essays Online Video Youtube

Sashk`O · 08.03.2014

sekk · 10.03.2014

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Dog-kaluga · 22.03.2014

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lyzer · 27.03.2014

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efka · 14.03.2014

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