Prove that every even degree polynomial function has a maximum or minimum in $\mathbb{R}$ 
Prove that every even degree polynomial function $f$ has maximum or minimum in $\mathbb{R}$. (without direct using of derivative and making $f'$)

The problem seems very easy and obvious but I don't know how to write it in a mathematical way.
For example if the largest coefficient is positive, it seem obvious to me that from a point $x=a$ to $+\infty$ the function must be completely ascending. And from $-\infty$ to a point $x=b$ the function must be completely descending. If it is not like that, its limit will not be $+\infty$ at $\pm\infty$. Now, because it is continuous, it will have a maximum and minimum in $[b,a]$ so it will have a minimum (because every other $f(x)$ where $x$ is outside $[b,a]$ is larger than $f(a)$ or $f(b)$ and we get the minimum that is less than or equal to both of them) . We can also the same for negative coefficient.
But I can't write this in a formal mathematical way.

In an even degree polynomial $p_{\,2n}(x) \quad |\; 0<n$, we have that
$$
\mathop {\lim }\limits_{x\; \to \, + \infty } p_{\,2n} (x) = \mathop {\lim }\limits_{x\; \to \, - \infty } p_{\,2n} (x) =  \pm \infty 
$$
and it does not have other poles other than those.
Assume that the limit is $+ \infty$.
Since a polynomial is a continuous surjective function,  its co-domain will be limited  below,  and  this limit will correspond to one or more
values of $x$.
The same applies, flipped over, if the limit is $-\infty$.
