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Question Number 189124 by Rupesh123 last updated on 12/Mar/23

Answered by witcher3 last updated on 15/Mar/23

$$\mathrm{x}+\mathrm{y}+\mathrm{x}⩾3\sqrt[3]{\mathrm{xyz}}$$$$\iff \mathrm{xyz}⩾3\sqrt[3]{\mathrm{xyz}};\mathrm{let}\mathrm{a}=\mathrm{xyz}$$$$\iff \mathrm{a}⩾3\sqrt[3]{\mathrm{a}}\Rightarrow {\mathrm{a}}^{\frac{2}{3}}⩾3\Rightarrow \mathrm{a}⩾\sqrt[2]{27}=3\sqrt{3}$$$$\mathrm{a}+\frac{1}{\mathrm{a}}=\mathrm{f}\left(\mathrm{a}\right)⩾2,\mathrm{\forall }\mathrm{a}>0$$$${\mathrm{f}}^{\prime }\left(\mathrm{a}\right)=1-\frac{1}{{\mathrm{a}}^2},\mathrm{\forall }\mathrm{a}>1\mathrm{f}\mathrm{is}\mathrm{increase}$$$$\mathrm{f}(3\sqrt{3)}=\sqrt{27}+\frac{1}{\sqrt{27}}=\frac{28}{\sqrt{27}}=\frac{28}{3\sqrt{3}}$$

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