I'm supposed to proof monotony. I have calculated the extremums first and realized that the function has a minimum and a maximum.
Is this enough to claim that the function isn't monotone because it has a minimum and a maximum? Or is a proof required to prove this statement?
This would safe me a lot time in the exam if you can just write that.
Let's take tasks like that:
"Proof if the function is monotone" or "Is the function monotone in $\mathbb{R}?$"
What do you think?
As several people have pointed out, this statement is false. The following statement is true: Let $I = (a,b)$ be an open interval, and suppose $f : I \to \mathbb{R}$ is monotone increasing and has a minimum $f(x_0) = m$ and a maximum $f(x_1) = M$. Then $f$ is identically $m$ on $(a,x_0)$ and identically $M$ on $(b,x_1)$. You can formulate an analogous statement for monotone decreasing functions by interchanging $x_0$ and $x_1$ in these two intervals on which $f$ is constant.
To prove this, just note that if $x \in (a,x_0)$, then monotone increasing implies $f(x) \leq f(x_0)$, but the fact that $x_0$ is a minimum implies $f(x) \geq f(x_0)$.
An implication of this statement phrased as your question title is that if a function which has a min and max on an open interval and is not constant on some "tail end" intervals cannot be monotone. As the comments have shown, these two additional hypotheses are both necessary. If the interval is not open, the min and max could occur at end points and the function need not be monotone in the middle. And if the min and max occur at tail ends, again the function need not be monotone in the middle.