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Show 6abc-b(a^2+c^2)<=(a^2+b^2)c+(b^2+c^2)a if a, b, c > 0

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uoor | Student, College Freshman | (Level 2) eNoter

Posted December 30, 2011 at 12:48 AM via web

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Show 6abc-b(a^2+c^2)<=(a^2+b^2)c+(b^2+c^2)a if a, b, c > 0

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted December 30, 2011 at 1:17 AM (Answer #1)

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I suggest you to move the term `b(a^2+c^2)`  to the right side such that:

`6abc =lt (a^2+b^2)c + (b^2+c^2)a + b(a^2+c^2)`

You need to use the following inequalities such that: `a^2+b^2gt=2ab ; a^2+c^2 gt= 2ac ; b^2+c^2 gt= 2bc` .

You need to multiply the inequality `a^2+b^2gt=2ab`  by the positive number c such that  `(a^2+b^2)cgt=2abc` .

You need to multiply the inequality `a^2+c^2gt=2ac`  by the positive number c such that  `(a^2+c^2)bgt=2abc` .

You need to multiply the inequality `c^2+b^2gt=2cb`  by the positive number a such that  `(c^2+b^2)agt=2abc` .

Adding the new inequalities yields:

`(a^2+b^2)c + (a^2+c^2)b + (c^2+b^2)agt=2abc + 2abc + 2abc`

`` `(a^2+b^2)c + (a^2+c^2)b + (c^2+b^2)agt=6abc`

The last line proves that the inequality `(a^2+b^2)c + (a^2+c^2)b + (c^2+b^2)agt=6abc`  is checked.

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