document.write( "Question 1177684: Let a and b be positive real numbers(That is a≥0, b≥0). Prove that a⁴+b⁴≥a³b+ab³
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Algebra.Com's Answer #806785 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "Let's consider this expression\r\n" );
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document.write( "    a^4 - a^3b + b^4 - ab^3.\r\n" );
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document.write( "Transform it this way  \r\n" );
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document.write( "    a^4 - a^3b + b^4 - ab^3 =  +  =  =  = .\r\n" );
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document.write( "So, our starting expression is the product of two quadratic polynomials\r\n" );
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document.write( "      and  .\r\n" );
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document.write( "They both are positively defined; in other words, they never take negative values.\r\n" );
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document.write( "Therefore,   >= 0  for all values of \"a\" and \"b\".\r\n" );
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document.write( "It implies that the original expression is never negative \r\n" );
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document.write( "    a^4 - a^3b + b^4 - ab^3 >= 0.\r\n" );
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document.write( "It means that\r\n" );
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document.write( "    a^4 + b^4 >= a^3b + ab^3,\r\n" );
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document.write( "which is what has to be proved.\r\n" );
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