Question 1102452
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After reading your post I still have no clear idea what really and exactly do you want !?


It is not the way to write/to present a mathematical request to the forum in such a form.


There is one simple rule which may help you in such deals and in the life:



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      ASK ONE AND ONLY ONE QUESTION at a time in your written request.
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Now, if you want to find  min f(x,y) = (x^2 + y^2) subject to xy >= 4,

then below is very helpful wording visualization/description:


<pre>
     the function f(x,y) = (x^2 + y^2)  represents a paraboloid in 3D space  {{{R^3}}}.

     It is a rotational paraboloid with the axis "Z" as the axis of rotation.


     Your task is to find its minimum over the domain xy >= 4 in the coordinate plane  (x,y)  (coordinate plane  Z = 0).


         In 3D,  the domain xy >= 4  is the cylindrical unbounded (in z-coordinate) 3D volume/body.

         When I say "cylindrical (in z-coordinate) 3D volume/body", I mean that this volume/body has vertical generating line 
         over the curve/the hyperbola  xy = 4  in  QI  and  QIII  quadrants in the plane Z = 0.


     If you imagine it MENTALLY  in your head, it will become clear to you, that the minimum you are looking for 

     is over the points  (2,2)  and  (-2,-2)  of the (x,y) plane.

     Exactly where the circle x^2 + y^2 = 8 or the radius {{{2*sqrt(2)}}}  centered at the origin of the coordinate plane 

     touches the hyperbola xy = 4 in the plane (x,y).


     And this minimum is equal to (2^2 + 2^2) = 4 + 4 = 8.
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