document.write( "Question 1180775: Find the maximum values of the function f(x,y,z)=x^2y^2z^2 subject to the constraint x^2+y^2+z^2=196.
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Algebra.Com's Answer #810564 by ikleyn(52835) $\"\"$ $\"About$

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document.write( "By analogy with the well known  AM-GM inequality (\"Arithmetic Mean - Geometric Mean inequality\") for two variables \"a\" and \"b\"\r\n" );
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document.write( "    ab <= ,         (1)\r\n" );
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document.write( "there is AM-GM inequality for three variables \"a\", \"b\" and \"c\"\r\n" );
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document.write( "    abc <= .      (2)\r\n" );
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document.write( "    Inequalities (1) and (2) are valid for any two and three variables, respectively, that are real non-negative numbers.\r\n" );
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document.write( "    In inequalities (1) and (2), equalities are achieved if and only if  a = b  (for (1))  or  a = b = c (for (2)).\r\n" );
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document.write( "Apply inequality (2), taking  \r\n" );
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document.write( "    a = x^2,  b = y^2,  c = z^2.\r\n" );
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document.write( "You will get\r\n" );
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document.write( "    x^2*y^2*z^2 <=  =  = .\r\n" );
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document.write( "Thus the maximum value of  x^2*y^2*z^2,  under the constraint  x^2+y^2+z^2 = 196  is   = .    ANSWER\r\n" );
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document.write( "It is achieved when  x^2 = y^2 = z^2 = ,  i.e.  x = y = z = +/-  = 8.082904  (rounded).\r\n" );
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document.write( "In all, there are 8 points on the 3D sphere surface  x^2 + y^ + z^2 = 196,\r\n" );
document.write( "where the maximum value of x^2*y^2*z^2 is achieved - one such point in each octant.\r\n" );
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