Problem with a polynomial and its second derivative

Question

I need to prove the following:

let p(x) be a polynomial with even degree and leading coefficient > 0, and p(x) - p''(x)>0 for all real x.

then p(x)>0 for all real x.

Even a partial help would be appreciated.

Answer 1

Such polynomial needs to have a global minimum.

Suppose there exists $y$ such that $p(y)\leq 0$, and call $x$ the minimum, because it's a global minimum, we must have $p''(x)\geq 0$ (see here for example) and $p(x)\leq 0$ (this is because $p(x)\leq p(y)$ ). This means that $p(x)-p''(x)\leq 0$, which is a contradiction.