$xy=1 \implies $minimum $x+y=$? If $x,y$ are real positive numbers such that $xy=1$, how can I find the minimum for $x+y$?

Consider a rectangle with dimension (x, y) of unit area, the half perimeter is defined as $a+b$
a square is a one and only one special form of rectangle with equal dimension should either be a maxima or minima
Now lets calculate the half perimeter of a square
$x.y = 1$
as $x = y$
we have
$x.x = 1$
so $x = 1$
which gives us 
$x + y = 2$
Consider any one rectangular figure
$x = 2$ 
and 
$y = \frac{1}{2}$
this gives the perimeter as
$x + y = 2 + 1/2 = 2.5$
$2.5(rectangle) > 2 (square)$
which makes us to believe
a square will have the minimum perimeter
thus we can conclude
the minimum value of $x + y$ is $2$
