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Question Number 149868 by mathdanisur last updated on 07/Aug/21

if a; b and c are the dimensions of a

cuboid with the diagonal d then prove

d ⩽ \sqrt{\frac{a^{3}}{b} + \frac{b^{3}}{c} + \frac{c^{3}}{a}}

Answered by dumitrel last updated on 08/Aug/21

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

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

\frac{c^{3}}{a} + a c \overset{a m - g m}{⩾} 2 c^{2} \Rightarrow

\Rightarrow \frac{a^{3}}{b} + \frac{b^{3}}{c} + \frac{c^{3}}{b} ⩾ 2 (a^{2} + b^{2} + c^{2}) - (a b + b c + a c) ⩾ a^{2} + b^{2} + c^{2} = d^{2}

Commented by mathdanisur last updated on 08/Aug/21

Thank You S er, cool