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 : /

Your expression for the derivative is wrong, but I'll let you sort that out. What is important is that:


*

*$f(0)=0$;

*there are just two values of $x$ for which $f'(x)=0$, and one of them is positive, the other negative;

*$f(x)$ tends to zero as $x$ tends to $\pm\infty$;

*$f(x)$ has the same sign as $x$.


These facts are enough to prove the nature of the stationary points. You should draw a graph if it helps. (You certainly don't have to calculate the second derivative.)
