Algebra.Com's Answer #717198 by ikleyn(52908)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "After reading your post I still have no clear idea what really and exactly do you want !?\r
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document.write( "It is not the way to write/to present a mathematical request to the forum in such a form.\r
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document.write( "There is one simple rule which may help you in such deals and in the life:\r
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document.write( " ASK ONE AND ONLY ONE QUESTION at a time in your written request.
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document.write( "Now, if you want to find min f(x,y) = (x^2 + y^2) subject to xy >= 4,\r
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document.write( "then below is very helpful wording visualization/description:\r
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document.write( " the function f(x,y) = (x^2 + y^2) represents a paraboloid in 3D space  .\r\n" );
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document.write( " It is a rotational paraboloid with the axis \"Z\" as the axis of rotation.\r\n" );
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document.write( " Your task is to find its minimum over the domain xy >= 4 in the coordinate plane (x,y) (coordinate plane Z = 0).\r\n" );
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document.write( " In 3D, the domain xy >= 4 is the cylindrical unbounded (in z-coordinate) 3D volume/body.\r\n" );
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document.write( " When I say \"cylindrical (in z-coordinate) 3D volume/body\", I mean that this volume/body has vertical generating line \r\n" );
document.write( " over the curve/the hyperbola xy = 4 in QI and QIII quadrants in the plane Z = 0.\r\n" );
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document.write( " If you imagine it MENTALLY in your head, it will become clear to you, that the minimum you are looking for \r\n" );
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document.write( " is over the points (2,2) and (-2,-2) of the (x,y) plane.\r\n" );
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document.write( " Exactly where the circle x^2 + y^2 = 8 or the radius  centered at the origin of the coordinate plane \r\n" );
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document.write( " touches the hyperbola xy = 4 in the plane (x,y).\r\n" );
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document.write( " And this minimum is equal to (2^2 + 2^2) = 4 + 4 = 8.\r\n" );
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