document.write( "Question 1119428: Without the derivative find the value of minima of
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Algebra.Com's Answer #734997 by ikleyn(52781) $\"\"$ $\"About$
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document.write( "Let me introduce new variable  u = x+2.5.\r\n" );
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document.write( "Then  x+2 = u-0.5,  x+3 = u+0.5  and  (x+2)*(x+3) = (u-0.5)*(u+0.5) = .\r\n" );
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document.write( "Similarly,  x+1 = u-1.5,  x+4 = u+1.5  and  (x+1)*(x+4) = (u-1.5)*(u+1.5) = .\r\n" );
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document.write( "And the product  (x+1)(x+2)(x+3)(x+4)  is  .\r\n" );
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document.write( "Thus we have two quantities  r =   and  s = ,  such that  their difference  (r - s)  has the constant value of 2\r\n" );
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document.write( "    r - s =  -  = 2,     (1)\r\n" );
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document.write( "and we are looking for the minimum value of the product  r*s.\r\n" );
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document.write( "From (1),  r = 2+s,  so we are looking for the minimum of the quadratic function \r\n" );
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document.write( "    r*s = (2+s)*s = .\r\n" );
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document.write( "The minimum is EXACTLY at the MIDPOINT BETWEEN the ZEROES  s= 0 and  s= -2, i.e. at s= -1.\r\n" );
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document.write( "s= -1 means  u^2 - 2.25 = -1,  or  u^2 = -1 + 2.25 = 1.25,\r\n" );
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document.write( "      which, in turn, means   = 1.25  ====>  x = -2.5 +- .\r\n" );
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document.write( "Thus we found two minimums for the original function  y(x) = (x+1)(x+2)(x+3)(x+4).\r\n" );
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document.write( "These minimums are   = -2.5 -  = -3.62 (approximately), and\r\n" );
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document.write( "                     = -2.5 +  = -1.38 (approximately).\r\n" );
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document.write( "The plot below visually confirms this answer.\r\n" );
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document.write( "Plot y = (x+1)(x+2)(x+3)(x+4)\r\n" );
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document.write( "The Table below (obtained with MS Excel) also confirms the answer.\r\n" );
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document.write( " x       y(x)\r\n" );
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document.write( "		\r\n" );
document.write( "-3.9    -0.4959		\r\n" );
document.write( "-3.8    -0.8064		\r\n" );
document.write( "-3.7    -0.9639		\r\n" );
document.write( "-3.6    -0.9984	  (*)	-3.618033989\r\n" );
document.write( "-3.5	-0.9375		\r\n" );
document.write( "-3.4	-0.8064		\r\n" );
document.write( "-3.3	-0.6279		\r\n" );
document.write( "-3.2	-0.4224		\r\n" );
document.write( "-3.1	-0.2079		\r\n" );
document.write( "-3	 0.0000		\r\n" );
document.write( "-2.9	 0.1881		\r\n" );
document.write( "-2.8	 0.3456		\r\n" );
document.write( "-2.7	 0.4641		\r\n" );
document.write( "-2.6	 0.5376		\r\n" );
document.write( "-2.5	 0.5625		\r\n" );
document.write( "-2.4	 0.5376		\r\n" );
document.write( "-2.3	 0.4641		\r\n" );
document.write( "-2.2	 0.3456		\r\n" );
document.write( "-2.1	 0.1881		\r\n" );
document.write( "-2	 0.0000		\r\n" );
document.write( "-1.9	-0.2079		\r\n" );
document.write( "-1.8	-0.4224		\r\n" );
document.write( "-1.7	-0.6279		\r\n" );
document.write( "-1.6	-0.8064		\r\n" );
document.write( "-1.5	-0.9375		\r\n" );
document.write( "-1.4	-0.9984	  (*)	-1.381966011\r\n" );
document.write( "-1.3	-0.9639		\r\n" );
document.write( "-1.2	-0.8064		\r\n" );
document.write( "-1.1	-0.4959		\r\n" );
document.write( "-1	 0.0000		\r\n" );
document.write( "-0.9	 0.7161		\r\n" );
document.write( "-0.8	 1.6896		\r\n" );
document.write( "-0.7	 2.9601		\r\n" );
document.write( "-0.6	 4.5696		\r\n" );
document.write( "-0.5	 6.5625		\r\n" );
document.write( "-0.4	 8.9856		\r\n" );
document.write( "-0.3	11.8881		\r\n" );
document.write( "-0.2	15.3216		\r\n" );
document.write( "-0.1	19.3401	\r\n" );
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document.write( "If you want to get the values of minimums, you can take them from the table.\r\n" );
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document.write( "These values both are equal to  -0.9984 (approximately).\r\n" );
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