Proof that a degree $4$ polynomial has a minimum 
Let $$f(x) = x^4+a_3x^3+a_2x^2+a_1x+a_0.$$ Prove that $f(x)$ has a minimum point in $\Bbb{R}$.

The Extreme Value Theorem implies that the minimum exists in some $[a,b]$, but how do I find the right $a$ and $b$?

If $x>|a_3|+\sqrt{|a_2|}+\sqrt[3]{|a_1|}+\sqrt[4]{|a_0|}$ then $f(x)>0$.
