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if-a-b-c-gt-0-and-ab-bc-ca-3-then-prove-that-a-3-1-b-3-1-c-3-1-1-3-2-




Question Number 156359 by MathSh last updated on 10/Oct/21
if  a;b;c>0  and  ab+bc+ca=3  then prove that:  (((a^3  + 1)(b^3  + 1)(c^3  + 1)))^(1/3)  ≥ 2
$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=\mathrm{3} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\sqrt[{\mathrm{3}}]{\left(\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{1}\right)\left(\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{1}\right)\left(\mathrm{c}^{\mathrm{3}} \:+\:\mathrm{1}\right)}\:\geqslant\:\mathrm{2} \\ $$

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