#### Chapter 7 Differential Calculus

Section 7.4 Properties of Functions

# 7.4.2 Monotony

The derivative of a function can be used to study the growth behaviour, i.e. whether the function values increase or decrease for increasing values of $x$. For this purpose, we consider a function $f:D\to ℝ$ that is differentiable on $\text{}\right]a;b\left[\text{}\subseteq D$:

If $f\text{'}\left(x\right)\le 0$ for all $x$ between $a$ and $b$, then $f$ is monotonically decreasing on the interval $\text{}\right]a;b\left[\text{}$.
If $f\text{'}\left(x\right)\ge 0$ for all $x$ between $a$ and $b$, then $f$ is monotonically increasing on the interval $\text{}\right]a;b\left[\text{}$.
Thus, it is sufficient to determine the sign of the derivative $f\text{'}$ to decide whether a function is monotonically increasing or decreasing on the interval $\text{}\right]a;b\left[\text{}$.
##### Example 7.4.1
The function $f:ℝ\to ℝ,x\to {x}^{3}$ is differentiable with $f\text{'}\left(x\right)=3{x}^{2}$. Since ${x}^{2}\ge 0$ for all $x\in ℝ$, we have $f\text{'}\left(x\right)\ge 0$, and therefore $f$ is monotonically increasing.
For $g:ℝ\to ℝ$ with $g\left(x\right)=2{x}^{3}+6{x}^{2}-18x+10$, the function $g\text{'}\left(x\right)=6{x}^{2}+12x-18=6\left(x+3\right)\left(x-1\right)$ 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.

$\begin{array}{cccc}\hfill x\hfill & \hfill x<-3\hfill & \hfill -3

For the function $h:ℝ\setminus \left\{0\right\}\to ℝ$ with $h\left(x\right)=\frac{1}{x}$, we have $h\text{'}\left(x\right)=-\frac{1}{{x}^{2}}$, that is $h\text{'}\left(x\right)<0$ 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 $h\left(-2\right)=-\frac{1}{2}$ and $h\left(1\right)=1$. Here, we have $-2<1$ but also $h\left(-2\right). 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 $\text{}\right]-\infty ;0\left[\text{}$ thus means that the restriction of $h$ on this interval is monotonically decreasing. Moreover, the function $h$ is also monotonically decreasing for all $x>0$.