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anemone
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If $a,\,b,\,c$ are positive numbers, show that $8(a^3+b^3+c^3)\ge (a+b)^3+(a+c)^3+(b+c)^3$.
anemone said:If $a,\,b,\,c$ are positive numbers, show that $8(a^3+b^3+c^3)\ge (a+b)^3+(a+c)^3+(b+c)^3----(1)$.
$a^3 + b^3 \geq a^2 b + b^2a$---(i)kaliprasad said:we have $a^3 + b^3 – a^2 b – b^2 a$
= $a^3 - a^2 b – b^2 a + b^3$
= $a^2(a-b)- b^2(a-b) = (a^2-b^2) (a-b) = (a+b)(a-b)^2> = 0$
Hence
$a^3 + b^3 > = a^2 b + b^2 a$
Multiply by 3 and add $a^3 + b^3$ on both sides
$4(a^3 + b^3) >= a^3 + b^3 + 3(a^2 b + b^2 a) > = (a+b)^3$
$4(a^3 + b^3) >= (a+b)^3$ .. (1)
Similarly
$4(b^3 + c^3) >= (b+c)^3$ ... (2)
$4(c^3 + a^3) >= (c+a)^3$ ...(3)
Adding (1), (2), (3) we get the result
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