document.write( "Question 1151382: The function $\"y=ax%5E3%2Bbx%5E2%2Bcx%2Bd\"$ has a relative maximum at the point (-2,27) and a relative minimum at the point (1,0). Find the values of a, b, c and d.\r
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Algebra.Com's Answer #773127 by ikleyn(52925) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "First derivative is\r\n" );
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document.write( "    y' = 3ax^2 + 2bx + c.\r\n" );
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document.write( "According to the condition, it has zeroes at x= -2 and x= 1.\r\n" );
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document.write( "Hence,   = -  = 1  and   = -2   \r\n" );
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document.write( "(according to Vieta's theorem).\r\n" );
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document.write( "From these equalities,\r\n" );
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document.write( "    b =   and  c = -6a.                (*)\r\n" );
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document.write( "Substitute these values to the equation for y.  You will get\r\n" );
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document.write( "    y =  +  -  + d.         (1)\r\n" );
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document.write( "Substitute x = -2  into (1)  to get y = 27.  It gives you\r\n" );
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document.write( "    27 = -8a + 6a + 12a + d,      or\r\n" );
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document.write( "    27 = 10a + d.                            (2)\r\n" );
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document.write( "Substitute x = 1  into (1)  to get y = 0.  It gives you\r\n" );
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document.write( "    0 = a +  - 6a  + d,      or\r\n" );
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document.write( "    0 = 2a + 3a - 12a + 2d,       or\r\n" );
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document.write( "    0 = -7a + 2d.                            (3)\r\n" );
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document.write( "Thus you have the system of 2 equations in 2 unknowns\r\n" );
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document.write( "    10a + d = 27,    (2')\r\n" );
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document.write( "    -7a + 2d = 0.    (3')\r\n" );
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document.write( "From (2'), express d = 27-10a  and substitute it into (3').  You will get\r\n" );
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document.write( "    -7a + 2*(27-10a) = 0,\r\n" );
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document.write( "    -7a + 54 - 20a   = 0\r\n" );
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document.write( "     54              = 7a + 20a\r\n" );
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document.write( "     54              = 27a\r\n" );
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document.write( "      a              = 54/27 = 2.\r\n" );
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document.write( "Thus  a = 2,  d = 27 - 10a = 27 - 20 = 7.\r\n" );
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document.write( "Finally,  from (*)  \r\n" );
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document.write( "     b =  =  = 3,   and \r\n" );
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document.write( "     c = -6a = -6*2 = -12.\r\n" );
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document.write( "ANSWER.  a = 2;  b = 3;  c = -12;  d = 7.\r\n" );
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