document.write( "Question 1047187: Find the maximum and minimum values of each given function and state the corresponding values of x . ( 0 <= x < 2pi )\r
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Algebra.Com's Answer #662696 by ikleyn(52786)\"\" \"About 
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\n" ); document.write( "Find the maximum and minimum values of each given function and state the corresponding values of x. ( 0 <= x < 2pi )\r
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document.write( "Take the derivative on \"x\": y'(x) = -sin(x) + 2*(-sin(x/2)*(1/2) = - sin(x) - sin(x/2).\r\n" );
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document.write( "Equate it to zero. You will get an equation\r\n" );
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document.write( "-sin(x) - sin(x/2) = 0,   or\r\n" );
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document.write( "2*sin(x/2)*cos(x/2) + sin(x/2) = 0,   or   ( after factoring )\r\n" );
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document.write( "sin(x/2)*(2*cos(x/2) + 1) = 0.\r\n" );
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document.write( "It deploys in two independent equations\r\n" );
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document.write( "1.  sin(x/2) = 0  --->  x/2 = 0, \"pi\", \"2pi\"  --->  x = 0  and  x = \"2pi\"  ( the only roots in the segment [\"0\",\"2pi\"] ).\r\n" );
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document.write( "2.  2*cos(x/2) + 1 = 0  --->  cos(x/2) = \"-1%2F2\"  ---> x/2 = \"2pi%2F3\", \"4pi%2F3\"  --->  x = \"4pi%2F3\"  ( the only root in the segment [\"0\",\"2pi\"] ).\r\n" );
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document.write( "Answer:\r\n" );
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document.write( "  a)  Maximum at x = 0:                        3.\r\n" );
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document.write( "  b)  Minimum at x = \"4pi%2F3\":  \"-1%2F2+%2B+2%2A%28-1%2F2%29\"   = -1.5.\r\n" );
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document.write( "  c)  Local maximum at x = \"2pi\": 1 + 2*(-1) = -1.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Plot y = cos(x)+2cos(x/2)\r
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