document.write( "Question 80864This question is from textbook CALCULUS
\n" ); document.write( ": help needed!
\n" ); document.write( "problem 16 on page 298.
\n" ); document.write( "f(x)= ax^3 + bx^2 + cx + d (a different from 0)
\n" ); document.write( "Find conditions on a, b, and c to ensure that f is always increasing or always decreasing on (negative infinitif, positive infinitif).\r
\n" ); document.write( "\n" ); document.write( "1 solution
\n" ); document.write( "Please do not contact:allmaiga@hotmail.com because I gave that email by mistake.
\n" ); document.write( "tanks. \n" ); document.write( "

Algebra.Com's Answer #58016 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
f(x)= ax^3 + bx^2 + cx + d (a different from 0)
\n" ); document.write( "Find conditions on a, b, and c to ensure that f is always increasing or always decreasing on (negative infinitif, positive infinitif).
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document.write( "The condition that f is always increasing or decreasing \r\n" );
document.write( "on (-oo,oo) is that f'(x) is either always positive, or\r\n" );
document.write( "always negative.\r\n" );
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document.write( "So we find the derivative f'(x)\r\n" );
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document.write( "f(x)= ax³ + bx² + cx + d\r\n" );
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document.write( "f'(x) = 3ax² + 2bx + c\r\n" );
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document.write( "Then set that > 0\r\n" );
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document.write( "3ax² + 2bx + c > 0\r\n" );
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document.write( "For this to be true, \r\n" );
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document.write( "f'(x) = 3ax² + 2bx + c \r\n" );
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document.write( "must represent a parabola\r\n" );
document.write( "which is always either above the x-axis\r\n" );
document.write( "or always below the x-axis.  This means\r\n" );
document.write( "that f'(x) can have no real zeros.\r\n" );
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document.write( "Therefore its discriminant must be\r\n" );
document.write( "negative.  The discriminant of\r\n" );
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document.write( "Ax² + Bx + C is B²-4AC, and in our case\r\n" );
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document.write( "A = 3a, B=2b, C = c, so the discriminant is\r\n" );
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document.write( "(2b)² - 4(3a)(c) or 4b² - 12ac, so we must have\r\n" );
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document.write( "4b² - 12ac < 0 or\r\n" );
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document.write( "       4b² < 12ac or\r\n" );
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document.write( "        b² < 3ac\r\n" );
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document.write( "is the requirement.\r\n" );
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document.write( "Now for a word of caution.  There is some disagreement\r\n" );
document.write( "among mathematicians as to whether to say that a \r\n" );
document.write( "function is increasing or decreasing at a horizontal \r\n" );
document.write( "inflection point.  If your teacher is one who \r\n" );
document.write( "says that the function f(x) = x³ + 3x² + 3x, graphed\r\n" );
document.write( "below\r\n" );
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document.write( "is increasing everywhere, even at the point (-1,-1), \r\n" );
document.write( "where it has a horizontal inflection point, i.e. its \r\n" );
document.write( "derivative is 0, indicated below by the horizontal \r\n" );
document.write( "tangent line:\r\n" );
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document.write( "    \r\n" );
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document.write( "then you must replace all the strict inequalities\r\n" );
document.write( "\" < \" by \" <, and then the requirement will\r\n" );
document.write( "be \r\n" );
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document.write( "        b² < 3ac\r\n" );
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document.write( "and also you would have to change the initial\r\n" );
document.write( "statement above to\r\n" );
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document.write( "The condition that f is always increasing or decreasing \r\n" );
document.write( "on (-oo,oo) is that f'(x) is either always nonpositive,\r\n" );
document.write( "or always nonnegative.\r\n" );
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document.write( "So be sure to ask your teacher whether or not he or she\r\n" );
document.write( "considers a function to be increasing (or decreasing)\r\n" );
document.write( "at a horizontal inflection point, where the derivative\r\n" );
document.write( "is 0, as long as it is increasing (or decreasing)\r\n" );
document.write( "everywhere else.\r\n" );
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document.write( "Edwin

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