Without using derivatives, prove that a polynomial of even degree either has a maximum or a minimum on $\displaystyle (-\infty,\infty).$ Give a simple criterion for deciding which it has.
If the leading coefficient of the polynomial $\displaystyle p(x)$ is positive then $\displaystyle p(x) \rightarrow \infty$ as $\displaystyle x \rightarrow \pm \infty$. If said coefficient is negative $\displaystyle p(x) \rightarrow -\infty$. So with an argument similar to the one used here you can conclude that in the first case you have a minimum and in the second a maximum.