document.write( "Question 1037029: Find the conditions on the real number \"p\" for which \"x%5E4+-4p%5E3x+%2B12+%3E0\" for all real numbers x . \n" ); document.write( "
Algebra.Com's Answer #651774 by Edwin McCravy(20081)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We want the conditions necessary so that the graph of\r\n" );
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document.write( "y = x4 - 4p3x + 12 \r\n" );
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document.write( "is always above the x-axis.\r\n" );
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document.write( "This is an even-degree polynomial with positive\r\n" );
document.write( "leading coefficient, thus its extreme right and\r\n" );
document.write( "left behavior is upward.  Polynomials are continuous\r\n" );
document.write( "at all points.  Thus its least relative minimum is \r\n" );
document.write( "its absolute minimum.  We find its relative extrema\r\n" );
document.write( "by setting its derivative = 0:\r\n" );
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document.write( "y' = 4x3-4p3 = 0\r\n" );
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document.write( "         4x3 = 4p3 \r\n" );
document.write( "           \r\n" );
document.write( "           x = p\r\n" );
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document.write( "So y has but one relative minimum at x = p,\r\n" );
document.write( "we substitute to find the y-coordinate of the \r\n" );
document.write( "relative (and absolute) minimum point:\r\n" );
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document.write( "y = x4 - 4p3x + 12\r\n" );
document.write( "y = p4 - 4p3p + 12\r\n" );
document.write( "y = p4 - 4p4 + 12\r\n" );
document.write( "y = -3p4+12\r\n" );
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document.write( "Since we want y to always be positive:\r\n" );
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document.write( "-3p4+12 > 0\r\n" );
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document.write( "-3p4 > -12\r\n" );
document.write( "  p4 < 4\r\n" );
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document.write( "Taking positive square roots of both sides:\r\n" );
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document.write( "   p2 < 2\r\n" );
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document.write( "Taking positive square roots of both sides again:\r\n" );
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document.write( "    p < √2\r\n" );
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document.write( "That's the restriction on p. \r\n" );
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document.write( "[It's interesting to note that p can be 0 or any negative\r\n" );
document.write( "value].\r\n" );
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document.write( "As a check, √2 is 1.414...\r\n" );
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document.write( "When p=1.42, just a smidgen above √2,\r\n" );
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document.write( "\"graph%28200%2C200%2C-2%2C3%2C-2%2C3%2Cx%5E4-4%281.42%29%5E3x%2B12%29\" \r\n" );
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document.write( "The graph's minimum point is a smidgen below the x-axis.  \r\n" );
document.write( "So y is NOT always positive.  But\r\n" );
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document.write( "when p=1.41, just a smidgen below √2, \r\n" );
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document.write( "\"graph%28200%2C200%2C-2%2C3%2C-2%2C3%2Cx%5E4-4%281.41%29%5E3x%2B12%29\" \r\n" );
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document.write( "Its minimum point is a smidgen above the x-axis\r\n" );
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document.write( "Edwin

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