I think I have a confusion with the Lagrange Theorem and its applications. But I think I also have a conceptual confusion with compact and non compact sets, which doesn't let me completely understand when a function attains both the max and min(Weierstrass Thm).
First, I would really appreciate if you could explain to me how to know if a set is compact (I know it has to be closed and bounded) but I would like to know what being closed really means. I read something like: if a set is closed then the complementary is open. If a set is open you "can put a ball inside" but if it is closed you can't. And do closed sets include the boundary itself? I don't really understand this, so I would really appreciate a clarification.
Second, when calculating extrema subject to a constraint: If my set is given by an equality I would apply Lagrange. My question is: to apply Lagrange, do I NECESSARILY need that the set is COMPACT?
I'm confused with this, because I thought I indeed needed this condition but then I saw an exercise that calculated the extrema on the following function USING Lagrange multipliers
f(x,y) = x^2+y^2 on D= {xy=1}
It got the critical points (1,1) and (-1,-1) and then, it just concluded that both those points were minimums.
I'm totally lost on how to arrive to this conclusion. How do I conclude that those points are minimums?
Thanks a lot for your help.
Closed sets include their own boundary. The boundary is the stuff that's between any closed set containing your set and not in any open set inside your set. It's the intersection of the closure of the set itself and the closure of the complement of the set.
One definition of closed sets that I find intuitive is that a set is closed iff it contains its own limit points.
When you apply Lagrange multipliers, you find all the critical points - those that in a certain sense are where the derivative is 0. These are not necessarily maxima or minima (ex: 0 in $x^3$).
The maxima/minima is either non-existent, on the boundary, or not on the boundary. All non boundary extremal points are critical points. You can consider those, and then restrict further by restricting to the boundary and optimizing.
The nice thing about continuous functions on compact spaces is that the maxima/minima will always exist. If it's not compact, there may be no maxima/minima. For example, the open the function $f(x)=x$ on the unit interval $(0,1)$ has no maxima or minima since there's always a point closer to 0/1 with a lower value for any point inside the interval. The closed/compact set $[0,1]$ however does attain maxima/minima at 0,1.