document.write( "Question 1019866: Which is the greatest/largest number x so that
\n" ); document.write( " $\"+%28a%5E2%2Ba%2B1%29%28b%5E2%2Bb%2B1%29%28c%5E2%2Bc%2B1%29+%3E=+xabc+\"$ \n" ); document.write( "

Algebra.Com's Answer #635892 by robertb(5830) $\"\"$ $\"About$

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The AM-GM inequality for three variables states that

for non-negative $\"x%5B1%5D\"$ , $\"x%5B2%5D\"$ , and $\"x%5B3%5D\"$ .
\n" ); document.write( "Hence

, or $\"a%5E2%2Ba%2B1%3E=3a\"$ (assuming a is non-negative).
\n" ); document.write( "Making the similar arguments for the variables b and c,
\n" ); document.write( "we get\r
\n" ); document.write( "\n" ); document.write( " $\"b%5E2%2Bb%2B1%3E=3b\"$ and $\"c%5E2%2Bc%2B1%3E=3c\"$ .\r
\n" ); document.write( "\n" ); document.write( "Thus, after multiplying corresponding sides of the three inequalities, we get\r
\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2%2Ba%2B1%29%2A%28b%5E2+%2B+b%2B1%29%2A%28c%5E2%2Bc%2B1%29%3E=27abc\"$ , and the greatest number x is 27. \n" ); document.write( "

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