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Question-219624

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Question Number 219624 by Nicholas666 last updated on 29/Apr/25
Answered by A5T last updated on 29/Apr/25
Σa(√(a^3 +15))≤(√(a^2 +b^2 +c^2 +d^2 ))(√(a^3 +b^3 +c^3 +d^3 +60)) ≤^((i)) (√(a^2 +b^2 +c^2 +d^2 ))(√(16(a^3 +b^3 +c^3 +d^3 ))) ⇒Σ(a^5 /( (√(a^3 +15))))=Σ(a^6 /(a(√(a^3 +15))))≥(((a^3 +b^3 +c^3 +d^3 )^2 )/(Σa(√(a^3 +15)))) ≥(((a^3 +b^3 +c^3 +d^3 )^2 )/( 4(√(a^3 +b^3 +c^3 +d^3 ))(√(a^2 +b^2 +c^2 +d^2 )))) ≥^((ii)) (((a^3 +b^3 +c^3 +d^3 )^2 )/(4(√(a^3 +b^3 +c^3 +d^3 ))×((2((a^3 +b^3 +c^3 +d^3 ))^(1/3) )/( (4)^(1/3) )))) =((4)^(1/3) /( 8))(a^3 +b^3 +c^3 +d^3 )^(7/6) ≥^((iii)) ((4)^(1/3) /8)×(4)^(7/6) =1 (i): a^3 +b^3 +c^3 +d^3 ≥4=4abcd ⇒60=60abcd≤15(a^3 +b^3 +c^3 +d^3 ) ⇒(√(a^3 +b^3 +c^3 +d^3 +60))≤(√((1+15)(a^3 +b^3 +c^3 +d^3 ))) (ii): Power Mean Inequality ⇒(((a^3 +b^3 +c^3 +d^3 )/4))^(1/3) ≥(((a^2 +b^2 +c^2 +d^2 )/4))^(1/2) ⇒(√(a^2 +b^2 +c^2 +d^2 ))≤((2((a^3 +b^3 +c^3 +d^3 ))^(1/3) )/( (4)^(1/3) )) (iii): (a^3 +b^3 +c^3 +d^3 )^(7/6) ≥(4(((abcd)^3 ))^(1/4) )^(7/6) =4^(7/6)
$$\Sigma\mathrm{a}\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{15}}\leqslant\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} }\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} +\mathrm{60}} \\ $$$$\overset{\left(\mathrm{i}\right)} {\leqslant}\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} }\sqrt{\mathrm{16}\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right)} \\ $$$$\Rightarrow\Sigma\frac{\mathrm{a}^{\mathrm{5}} }{\:\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{15}}}=\Sigma\frac{\mathrm{a}^{\mathrm{6}} }{\mathrm{a}\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{15}}}\geqslant\frac{\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right)^{\mathrm{2}} }{\Sigma\mathrm{a}\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{15}}} \\ $$$$\geqslant\frac{\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right)^{\mathrm{2}} }{\:\mathrm{4}\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} }\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} }} \\ $$$$\overset{\left(\mathrm{ii}\right)} {\geqslant}\frac{\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right)^{\mathrm{2}} }{\mathrm{4}\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} }×\frac{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} }}{\:\sqrt[{\mathrm{3}}]{\mathrm{4}}}} \\ $$$$=\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\:\mathrm{8}}\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right)^{\frac{\mathrm{7}}{\mathrm{6}}} \overset{\left(\mathrm{iii}\right)} {\geqslant}\frac{\sqrt[{\mathrm{3}}]{\mathrm{4}}}{\mathrm{8}}×\left(\mathrm{4}\right)^{\frac{\mathrm{7}}{\mathrm{6}}} =\mathrm{1} \\ $$$$ \\ $$$$\left(\mathrm{i}\right):\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \geqslant\mathrm{4}=\mathrm{4abcd} \\ $$$$\Rightarrow\mathrm{60}=\mathrm{60abcd}\leqslant\mathrm{15}\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right) \\ $$$$\Rightarrow\sqrt{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} +\mathrm{60}}\leqslant\sqrt{\left(\mathrm{1}+\mathrm{15}\right)\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right)} \\ $$$$ \\ $$$$\left(\mathrm{ii}\right):\:\mathrm{Power}\:\mathrm{Mean}\:\mathrm{Inequality} \\ $$$$\Rightarrow\left(\frac{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} }{\mathrm{4}}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \geqslant\left(\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} }{\mathrm{4}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\Rightarrow\sqrt{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} }\leqslant\frac{\mathrm{2}\sqrt[{\mathrm{3}}]{\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} }}{\:\sqrt[{\mathrm{3}}]{\mathrm{4}}} \\ $$$$ \\ $$$$\left(\mathrm{iii}\right):\:\left(\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} +\mathrm{d}^{\mathrm{3}} \right)^{\frac{\mathrm{7}}{\mathrm{6}}} \geqslant\left(\mathrm{4}\sqrt[{\mathrm{4}}]{\left(\mathrm{abcd}\right)^{\mathrm{3}} }\right)^{\frac{\mathrm{7}}{\mathrm{6}}} =\mathrm{4}^{\frac{\mathrm{7}}{\mathrm{6}}} \\ $$

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