Let-x-y-z-gt-0-x-y-z-3-Prove-That-1-x-2-2x-1-z-2-2z-3-1-y-2-y-9-x-y-z-24-1-3-3-17-3-3-

Tinku Tara October 5, 2023 Mensuration 0 Comments

Question Number 197950 by York12 last updated on 05/Oct/23

Let x,y,z>0 , x+y+z=3 Prove That : (1/( (√(x^2 +2x))))+(1/( (√(z^2 +2z))))+(√3)((1/(y+2))−(y/9))+((((√x)+(√y)+(√z)+24))^(1/3) /( (√3)))≥((17)/(3(√3)))

$${Let}\:{x},{y},{z}>\mathrm{0}\:,\:{x}+{y}+{z}=\mathrm{3}\:{Prove}\:{That}\:: \\ $$$$\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}}}+\frac{\mathrm{1}}{\:\sqrt{{z}^{\mathrm{2}} +\mathrm{2}{z}}}+\sqrt{\mathrm{3}}\left(\frac{\mathrm{1}}{{y}+\mathrm{2}}−\frac{{y}}{\mathrm{9}}\right)+\frac{\sqrt[{\mathrm{3}}]{\sqrt{{x}}+\sqrt{{y}}+\sqrt{{z}}+\mathrm{24}}}{\:\sqrt{\mathrm{3}}}\geqslant\frac{\mathrm{17}}{\mathrm{3}\sqrt{\mathrm{3}}} \\ $$

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Let-x-y-z-gt-0-x-y-z-3-Prove-That-1-x-2-2x-1-z-2-2z-3-1-y-2-y-9-x-y-z-24-1-3-3-17-3-3-

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