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)))


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

$L e t x, y, z > 0, x + y + z = 3 P r o v e T h a t :$ $\frac{1}{\sqrt{x^{2} + 2 x}} + \frac{1}{\sqrt{z^{2} + 2 z}} + \sqrt{3} (\frac{1}{y + 2} - \frac{y}{9}) + \frac{\sqrt[3]{\sqrt{x} + \sqrt{y} + \sqrt{z} + 24}}{\sqrt{3}} ⩾ \frac{17}{3 \sqrt{3}}$
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