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Question Number 40375 by behi83417@gmail.com last updated on 21/Jul/18

Answered by MJS last updated on 21/Jul/18

$$xyz(x+y+z)=1\Rightarrow z=-\frac{x+y}{2}\pm \frac{\sqrt{xy{(x+y)}^2+4}}{2\sqrt{xy}}\Rightarrow$$$$\Rightarrow p(x,y,z)=\frac{(x+y)(x^2{y}^2+1)}{xy}$$$$\mathrm{after}\mathrm{drawing}\mathrm{lots}\mathrm{of}\mathrm{graphs}\mathrm{I}\mathrm{cane}\mathrm{to}\mathrm{the}$$$$\mathrm{conclusion}\mathrm{that}\mathrm{this}\mathrm{has}\mathrm{a}\mathrm{minimum}\mathrm{with}$$$$y=x\Rightarrow z=x\pm \frac{\sqrt{x^4+1}}{x}\Rightarrow p(x,y,z)=\frac{2(x^4+1)}{x}$$$$p^{\prime }\left(x\right)=\frac{2(3x^4-1)}{x^2}=0\Rightarrow x=y=z=\frac{\sqrt{3\sqrt{3}}}{3}\approx .759836$$$$p(\frac{\sqrt{3\sqrt{3}}}{3},\frac{\sqrt{3\sqrt{3}}}{3},\frac{\sqrt{3\sqrt{3}}}{3})=\frac{8\sqrt[4]{3}}{3}\approx 3.50953$$$$$$

Commented by behi83417@gmail.com last updated on 21/Jul/18

$$thankyousomuchdear.$$

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