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How is 2(a^3+b^3+c^3)>=ab(a+b)+(b+c)bc+(a+c)ac?a>0 b>0 c>0

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rouche | Student, Undergraduate | (Level 1) Honors

Posted December 31, 2011 at 10:30 PM via web

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How is 2(a^3+b^3+c^3)>=ab(a+b)+(b+c)bc+(a+c)ac?

a>0

b>0

c>0

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 31, 2011 at 11:13 PM (Answer #1)

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You need to use the next inequality `(a-b)^2 gt= 0` .

Expanding the binomial yields: `a^2 - 2ab - b^2 gt= 0`

Hence, `a^2 - ab - ab- b^2 gt= 0 =gt a^2 - ab - b^2 gt= ab`

Multiplying both sides by (a+b) yields:

`(a+b)(a^2 - ab - b^2) gt= (a+b)ab`

The product from the left side yields the sum of cubes: `a^3 + b^3` .

`a^3 + b^3 gt= (a+b)ab`

You need to follow the same procedure for b^3 + c^3 and `a^3 + c^3 ` such that:

`b^3 + c^3 gt= (b+c)bc`

`` `a^3 + c^3 gt= (a+c)ac`

Adding these three inequalities yields:

`a^3 + b^3 + b^3 + c^3 + a^3 + c^3 gt= (a+b)ab + (b+c)bc + (a+c)ac`

`2(a^3+b^3+c^3) gt= (a+b)ab + (b+c)bc + (a+c)ac`

The last line proves the inequality

`2(a^3+b^3+c^3) gt= (a+b)ab + (b+c)bc + (a+c)ac`

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