7.4.4 Exercises

The monotony behaviour is determined by the derivative $f\text{'}$ of the function $f$. Since the graph of the derivative $f\text{'}$ is given in the exercise, we only have to read off on which intervals the graph lies above (or below) the $x$-axis: On the intervals $[-4.5;-4[$, $]0;3[$, and $]3;4[$, we have $f\text{'}(x)>0$, and hence, the function $f$ is monotonically increasing there. However, on the interval $]-4;0[$, we have $f\text{'}(x)<0$, and hence the function $f$ is monotonically decreasing there.
At an extremal point $x_e$ (maximum or minimum point) of a function $f$ (which does not lie on the boundary of the domain) the first derivative is zero: $f\text{'}(x_e)=0$. Graphically, this means that the tangent line to the graph of $f$ is a horizontal line. According to the graph, the zeros of $f\text{'}(x)$ are at $x_1=-4$, $x_2=0$, and $x_3=3$. Since $f$ is monotonically increasing on $]-4.5;-4[$ and monotonically decreasing on $]-4;0[$, $x_1=-4$ is a maximum point. This tells us that the function has a minimum point at $x_2=0$. (At $x_3=3$ the function has a saddle point.)