Find max/min of the following function Find the minimum e max distance (in $R^2$)
) between the point $Q =
(
3/
2
, −
3/
2
)$
and the set
$$B = \{(x, y) ∈ R^2
: yx = 1, x ≥ 0, y ≥ 0\}$$
In other words I have to find max /min points of the function 
$${(x-3/2)^2 + (y+3/2)^2}$$ 
The set $B$ is clearly neither bounded nor convex. If I use Lagrange I can only find local min/max, but how do I show they are global? In a previous question :Does the following function admit a maximum? you suggested me to maximize/minimize the $x$ component and the $y$ component independently, but it is not clear to me if I can do it in this exercise as well. It seens to me that choosing $x=3/2$, which clearly minimizes the first component of the sum, would restrict the choice of $y$ as $yx = 1 $.
Is there any way to show that the point found using lagrange is a global min? Possibly without using the bordered hessian method?
 A: (Note that you have a typo in your function, if the point is $(3/2,-3/2)$ then the second minus in your function should be a plus)
Using Lagrange multipliers is overkill here. In any case, Lagrange multipliers will give you critical points, which then you can check to see if they are the actual max/min. 
Here, $x,y>0$ (otherwise $xy\ne1$), and you have that $y=1/x$. So your function becomes 
$$
\left(x-\tfrac32\right)^2+\left(\tfrac1x+\tfrac32\right)^2,\ \ \ \ x>0. 
$$
Now you can do this using one-variable calculus.
A: Hint:
The  square of the distance function that you want minimize is $f(x,y)=(x-\frac{3}{2})^2+(y+\frac{3}{2})^2$ ( it seems that you have a wrong sign)
with the condition $g(xy)=xy=1$
so, using Lagrange multipliers, you have to solve;
$$
\begin{cases}
\nabla f=\lambda \nabla g\\
xy=1
\end{cases}
$$
can you do this?
A: Lets say y->0 and x->$\infty$ then value of above expression -> $\infty$ (so yeah no maximum).
$(x-\frac{3}{2})^2 +(y+\frac{3}{2})^2$(You have made a mistake here or given the point wrong) = $x^2 +y^2 +\frac{9}{2} + 3(y-x) = z^2 +3z +\frac{13}{2}$.
(as $x=\frac{1}{y}$, $(y-\frac{1}{y})$=z, z can be any real number , the quadratic equation have its minimum at $\frac{-3}{2}$ which is the required answer)
So minimum value is $\sqrt{\frac{17}{4}}$
