if-a-b-c-gt-0-then-9abc-a-3-b-3-c-3-a-2-bc-b-2-ca-c-2-ab-6-

Question Number 144083 by mathdanisur last updated on 21/Jun/21

i f a; b; c > 0 t h e n :

\frac{9 a b c}{a^{3} + b^{3} + c^{3}} + \frac{a^{2}}{b c} + \frac{b^{2}}{c a} + \frac{c^{2}}{a b} ⩾ 6

Answered by mitica last updated on 21/Jun/21

\frac{9 a b c}{a^{3} + b^{3} + c^{3}} + \frac{a^{2}}{b c} + \frac{b^{2}}{a c} + \frac{c^{2}}{a b} = \frac{9 a b c}{a^{3} + b^{3} + c^{3}} + \frac{a^{3} + b^{3} + c^{3}}{a b c} \overset{a m - g m}{⩾}

2 \sqrt{\frac{9 a b c}{a^{3} + b^{3} + c^{3}} \cdot \frac{a^{3} + b^{3} + c^{3}}{a b c}} = 2 \cdot 3 = 6

Commented by mathdanisur last updated on 21/Jun/21

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