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let-a-b-0-and-a-b-ab-3-show-that-38-55-1-a-2-2-1-b-2-2-1-a-2-b-2-1-1-463-812-1-a-3-2-1-

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Question Number 221399 by Nicholas666 last updated on 02/Jun/25
let a,b ≥ 0 and a + b + ab = 3 show that; ((38)/(55)) ≤ (1/(a^2 + 2)) + (1/(b^2 +2)) + (1/(a^2 + b^2 + 1)) ≤ 1 , ((463)/(812)) ≤ (1/(a^3 + 2)) + (1/(b^3 + 2)) + (1/(a^3 + b^3 + 1)) ≤ 1 , ((193)/(308)) ≤ (1/(a^2 + 2)) + (1/(b^3 + 2)) + (1/(a^3 + b^2 + 1)) ≤1 , and ((463)/(812)) ≤ (1/(a^2 + 2)) + (1/(b^3 + 2)) + (1/(a^2 + b^3 + 1)) ≤ ((11)/(10))
$$ \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{let}}\:\boldsymbol{{a}},\boldsymbol{{b}}\:\geqslant\:\mathrm{0}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{{a}}\:+\:\boldsymbol{{b}}\:+\:\boldsymbol{{ab}}\:=\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}; \\ $$$$\:\:\frac{\mathrm{38}}{\mathrm{55}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{2}} \:+\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\boldsymbol{{b}}^{\mathrm{2}} \:+\:\mathrm{1}}\:\leqslant\:\mathrm{1}\:\:,\:\:\:\:\:\:\: \\ $$$$\:\:\frac{\mathrm{463}}{\mathrm{812}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{3}} \:+\:\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{1}}\:\leqslant\:\mathrm{1}\:\:\:,\:\:\:\:\:\:\:\: \\ $$$$\:\:\frac{\mathrm{193}}{\mathrm{308}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{3}} \:+\:\boldsymbol{{b}}^{\mathrm{2}} \:+\:\mathrm{1}}\:\:\:\leqslant\mathrm{1}\:\:,\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{and}} \\ $$$$\:\:\frac{\mathrm{463}}{\mathrm{812}}\:\leqslant\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\boldsymbol{{a}}^{\mathrm{2}} \:+\:\boldsymbol{{b}}^{\mathrm{3}} \:+\:\mathrm{1}}\:\leqslant\:\frac{\mathrm{11}}{\mathrm{10}}\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

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