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?
Do you mean the function which asymptoticaly aproaches the plane when x ->infinity? 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)