How can I know whether the point is a maximum or minimum without much calculation? 
Find the maximum and minimum of this function and state whether they
  are local or global:
$$f: \mathbb{R} \ni x \mapsto \frac{x}{x^{2}+x+1} \in \mathbb{R}$$

\begin{align*} 
f'(x)&= \frac{-x^{2}+x}{\left(x^{2}+x+1\right)^{2}}\\
f'(x)&=0 \iff \frac{-x^{2}+x}{\left(x^{2}+x+1\right)^{2}}=0 \iff -x^2+x=0 \iff x(1-x)=0, \end{align*}
which gives $x_{1}=0, x_{2}=1$. Here comes the disturbing part, we need to know if these are maximum or minimum and for this we usually used the second derivative. But this would be soo exhausting, I don't even  want think of doing it. There must be an easier way and I remember someone here has even recommended me using monotony somehow.
But how can we do this here? Please do tell me, at home I got enough time to use second derivative but surely not in the exam : /

Another way that sometimes works is to compare the values of the function at these critical points. Often you can do this without the need to evaluate the function (ok, here the evaluation is immediate) $$f(x_1)=\frac{0}{0^2+0+1}=0$$ and $$f(x_2)=\frac{1}{1^2-1+1}=1$$ So, certainly $f(x_1)>f(x_2)$ and hence in $x_1$ you have a $\max$ and in $x_2$ a $\min$ (be careful for saddle points). Now, check what happens at $\pm \infty$ (take the limit) to conclude whether the $\min$ in $x_2$ and the $\max$ in $x_1$ are local or global. Spoiler:

 they are global.

