Question 1111386
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<pre>
x + y = {{{(sqrt(x))^2}}} + {{{(sqrt(y))^2}}} = {{{(sqrt(x) - sqrt(y))^2}}} + {{{2*sqrt(xy)}}}.


This line is an IDENTITY.


Now, the term  {{{(sqrt(x) - sqrt(y))^2}}}  is ALWAYS positive (non-negative), therefore, we can continue and transform this line / identity in THIS WAY


x + y = {{{(sqrt(x))^2}}} + {{{(sqrt(y))^2}}} = {{{(sqrt(x) - sqrt(y))^2}}} + {{{2*sqrt(xy)}}}  > =  {{{2*sqrt(xy)}}} = {{{2*sqrt(9)}}} = 2*3 = 6.


Thus we have established this inequality


x + y >= 6.


Taking x= y= 3,  we get the exact value  of 6  for  x+y.  


In all other cases we have inequality  x + y > 6.


So, under the given condition  xy >= 9,  the smallest value for  x + y  is  6.
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SOLVED.