Every real polynomial of two variables defined on the real plane

is this supposed to be remarkable?

>>8577963
It's remarkably false.

>>8577968
Proofs?

>>8577969
(1-xy)^2 + x^2

>>8577969
Bounded from below just means that the surface never goes below that point.

The surface could still approach that point arbitrarily closely and thus never have a minimum.

>>8577973
O, yea. Is there some variation on OPs claim? Or he just said stupid shit?

Every subspace of a completely metrizable space is completely metrizable.

>>8577983
Every such function on a bounded subset of R2 will achieve its minimum.

>>8577983
>Is there some variation on OPs claim?

Do you mean for it to be right? Well, yes. It is true for real polynomials of one variable.

Babby proof:

The only polynomials to be bounded from below are even degree polynomials.

Every (and only) even degree polynomials are either bounded from below or above.

Even degree polynomials have either minima or maxima, never both.

Consider some arbitrary even degree polynomial that is bounded from below.

Take its derivative, which yields an odd degree polynomial.

Every odd degree polynomial has at least one real root.

These roots are either maxima or minima. But they cannot be maximum because if an even polynomial has a maximum then all of its critical points would be maxima and if that were the case then one of them would have to be a global maximum but then that global maximum would imply that the function is bounded from above.

But we already supposed the function was bounded from below, and even polynomials cannot be bounded from both sides.

So all of those critical points must be minima and therefore one of them must be global, so such polynomials always attain their minimum.

>>8578007
no, every closed subset :)

>>8577970
Has a minimum at (0, 0).

>>8577992
Sure thing bro

>>8578013
>But they cannot be maximum because if an even polynomial has a maximum then all of its critical points would be maxima
that's not true, and even degree polynomials can have local maxima (ex: (x-1)(x-2)(x-3)(x-4))
What you need to say is precisely that if a nonconstant polynomial has at least two critical points, then they can't all be maxima or minima (which is pretty clear if you draw the situation)

>>8578057
take x close to 0 and y=1/x faggot, the function approaches 0 but never reaches this value

>>8577957
x-y

>>8578375
Is it a polynomial? Dumbass

>>8578041
The set A = [0, infinity) x [0, infinity) is closed yet the function f with f(x,y) = -x for all (x,y) in A has no minimum.

Let X denote a compact and convex subset of euclidean space R^n. This implies that the set of its extreme points is closed.

>>8578423
You can't read

>stealing fake theorems from a fake math page on Facebook

heck off dude

>>8578375
>>8578057
I messed up my math. You are quite right; (1-xy)^2 + x^2 has an infimum of zero, but no minimum.