Menu Close

Question-159494




Question Number 159494 by HongKing last updated on 17/Nov/21
Answered by qaz last updated on 18/Nov/21
(1+a)(1+(b/a))(1+(c/b))(1+((81)/c))  ≥(1+(√a)∙(√(b/a)))^2 (1+(√(c/b))∙(√((81)/c)))^2   =(1+(√b))^2 (1+(√((81)/b)))^2   ≥(1+(b)^(1/4) ∙(((81)/b))^(1/4) )^4   =(1+3)^4   =256
$$\left(\mathrm{1}+\mathrm{a}\right)\left(\mathrm{1}+\frac{\mathrm{b}}{\mathrm{a}}\right)\left(\mathrm{1}+\frac{\mathrm{c}}{\mathrm{b}}\right)\left(\mathrm{1}+\frac{\mathrm{81}}{\mathrm{c}}\right) \\ $$$$\geqslant\left(\mathrm{1}+\sqrt{\mathrm{a}}\centerdot\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}\right)^{\mathrm{2}} \left(\mathrm{1}+\sqrt{\frac{\mathrm{c}}{\mathrm{b}}}\centerdot\sqrt{\frac{\mathrm{81}}{\mathrm{c}}}\right)^{\mathrm{2}} \\ $$$$=\left(\mathrm{1}+\sqrt{\mathrm{b}}\right)^{\mathrm{2}} \left(\mathrm{1}+\sqrt{\frac{\mathrm{81}}{\mathrm{b}}}\right)^{\mathrm{2}} \\ $$$$\geqslant\left(\mathrm{1}+\sqrt[{\mathrm{4}}]{\mathrm{b}}\centerdot\sqrt[{\mathrm{4}}]{\frac{\mathrm{81}}{\mathrm{b}}}\right)^{\mathrm{4}} \\ $$$$=\left(\mathrm{1}+\mathrm{3}\right)^{\mathrm{4}} \\ $$$$=\mathrm{256} \\ $$
Commented by HongKing last updated on 18/Nov/21
thank you my dear Ser cool
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Ser}\:\mathrm{cool} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *