Positive polynomial in two real variables
Does every positive polynomial in two real variables attain its lower bound in the plane? :yuck:

Let's start by investigating how it could fail.
Do you know of any way that a continuous function can fail to attain its lower bound? 
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It seems to me that, according the Sylvester's theorem the positive defined polynomial never reaches the plane, and it does not matter how behave the continuous function. 
I was suggesting a possible line of attack: examime what properties a continuous function must have if it doesn't attain its lower bound, then prove a polynomial can't have those polynomials.
But it sounds like you already have a line of attack... how are you proposing to use Sylvester's theorem? (I don't recall the theorem; a quick google search doesn't provide anything that seems relevant) 
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