prove that p(x) is always nonnegative 
If $p(x)$ is a polynomial with a positive leading coefficient and even degree such that $p(x)\ge p''(x)$ for all real x, then $p(x)\ge 0$ for all real x.

I tried considering the case where $p(x) < 0$ for some x and limiting behaviour. $p'(x)$ is eventually negative for small enough x and eventually positive for large enough x. Also, p is eventually positive for all x with large enough absolute value. I was thinking of applying the Intermediate value theorem somehow, but I can't seem to get a contradiction.

A real polynomial with even degree and positive leading coefficients attains its minimum at some point $x_0$, and at a minimum is $p''(x_0) \ge 0$. It follows that
$$
 p(x) \ge p(x_0) \ge p''(x_0) \ge 0
$$
for all $x \in \Bbb R$.
