how-to-prove-that-3d-3-4d-2-3d-1-2-5-d-1-2-d-2-2-d-3-2-d-2-d-1-2-d-3-d-2-2-d-1-d-2-d-3-2-

Question Number 201477 by York12 last updated on 07/Dec/23

how to prove that

{({3 d}_{3} + {4 d}_{2} + {3 d}_{1})}^{2} ⩽ 5 (d_{1}^{2} + d_{2}^{2} + d_{3}^{2} + {(d_{2} + d_{1})}^{2} + {(d_{3} + d_{2})}^{2} + {(d_{1} + d_{2} + d_{3})}^{2})

Answered by AST last updated on 07/Dec/23

L e t d_{1} = a; d_{2} = b; d_{3} = c

5 (3 a^{2} + 4 b^{2} + 3 c^{2} + 4 a b + 4 b c + 2 c a) \overset{?}{⩾}

9 a^{2} + 16 b^{2} + 9 c^{2} + 24 b c + 24 a b + 18 a c

\Leftrightarrow 3 a^{2} + 2 b^{2} + 3 c^{2} ⩾ 2 a b + 2 b c + 4 a c

a^{2} + b^{2} ⩾ 2 a b; 2 a^{2} + 2 c^{2} ⩾ 4 a c; b^{2} + c^{2} ⩾ 2 b c

\Rightarrow 3 a^{2} + 2 b^{2} + c^{2} ⩾ 2 a b + 2 b c + 4 a c \Rightarrow o r i g i n a l i n e q u a l i t y

i s t r u e .

Commented by York12 last updated on 07/Dec/23

thanks sir

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