Show that if $p(x)$ is a polynomial, $|p(x)|$ attains its minimum. 
Show that if $p(x)$ is a polynomial, $|p(x)|$ attains its minimum.

Attempt
Let $p(x) = a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Then if $|p(x)| =  |a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0|$. If $x > 0,$ then $|p(x)| = |a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0| \leq |a_n|x^n+|a_{n-1}|x^{n-1}+\cdots+|a_1|x+|a_0|.$ Then we can take there derivative of $|p(x)|$ to get a bound to find critical points. I am not sure how to deal with the case of $x \leq 0$.

Does $|p(x)|$ always attain $p(x)$'s minimum?
Try using a $p(x)$ such that $a_0<0$ and $a_0$ happens to be its minimum [example: $p(x)=x^2-1$]. Then $|p(x)|$ will never attain $a_0$, but $|a_0|$.
Maybe I'm misunderstanding the question. Try to word it better.
