7.4.2 Monotony

The function $f:ℝ\to ℝ,x\to x^3$ is differentiable with $f\text{'}(x)=3x^2$. Since $x^2\ge 0$ for all $x\in ℝ$, we have $f\text{'}(x)\ge 0$, and therefore $f$ is monotonically increasing.
For $g:ℝ\to ℝ$ with $g(x)=2x^3+6x^2-18x+10$, the function $g\text{'}(x)=6x^2+12x-18=6(x+3)(x-1)$ has the roots $x_1=-3$ and $x_2=1$. If the monotony of the function $g$ is investigated, then three regions are to be distinguished in which $g\text{'}$ has a different sign.
The following table is used to determine in which region the derivative of $g$ is positive or negative. These regions correspond to the monotony regions of $g$. The entry "$+$" says that the considered term is positive on the given interval. If the term is negative, then "$-$" is entered.

${\displaystyle \begin{array}{cccc}\hfill x\hfill & \hfill x<-3\hfill & \hfill -3<x<1\hfill & \hfill 1<x\hfill \\ \\ \hfill x+3\hfill & \hfill -\hfill & \hfill +\hfill & \hfill +\hfill \\ \hfill x-1\hfill & \hfill -\hfill & \hfill -\hfill & \hfill +\hfill \\ \hfill g\text{'}(x)\hfill & \hfill +\hfill & \hfill -\hfill & \hfill +\hfill \\ \\ \hfill g\mathrm{\hspace{0.5em}\hspace{0.5em}}\text{is monotonically}\mathrm{\hspace{0.5em}\hspace{0.5em}}\hfill & \hfill \mathrm{\hspace{0.5em}\hspace{0.5em}}\text{increasing}\mathrm{\hspace{0.5em}\hspace{0.5em}}\hfill & \hfill \mathrm{\hspace{0.5em}\hspace{0.5em}}\text{decreasing}\mathrm{\hspace{0.5em}\hspace{0.5em}}\hfill & \hfill \mathrm{\hspace{0.5em}\hspace{0.5em}}\text{increasing}\mathrm{\hspace{0.5em}\hspace{0.5em}}\hfill \end{array}}$
For the function $h:ℝ\setminus \{0\}\to ℝ$ with $h(x)=\frac{1}{x}$, we have $h\text{'}(x)=-\frac{1}{x^2}$, that is $h\text{'}(x)<0$ for all $x\ne 0$.
Even though the function $h$ exhibits the same monotony behaviour for the two subregions $x<0$ and $x>0$, it is not monotonically decreasing on the entire region. As a counterexample let us consider the function values $h(-2)=-\frac{1}{2}$ and $h(1)=1$. Here, we have $-2<1$ but also $h(-2)<h(1)$. This corresponds to an increasing growth behaviour if we change from one subregion to the other. The statement that the function $h$ is monotonically decreasing on $]-\infty ;0[$ thus means that the restriction of $h$ on this interval is monotonically decreasing. Moreover, the function $h$ is also monotonically decreasing for all $x>0$.