#### Chapter 7 Differential Calculus

Section 7.4 Properties of Functions

# 7.4.4 Exercises

##### Exercise 7.4.5
Specify the (maximum) open intervals on which the function $f$ with $f\left(x\right):=\frac{{x}^{2}-1}{{x}^{2}+1}$ is monotonically increasing or decreasing.
• $f$ is monotonically
on $\text{}\right]-\infty ;0\left[\text{}$.
• $f$ is monotonically
on $\text{}\right]0;\infty \left[\text{}$.

##### Exercise 7.4.6
Specify the (maximum) open intervals $\text{}\right]c;d\left[\text{}$ on which the function $f$ with $f\left(x\right):=\frac{{x}^{2}-1}{{x}^{2}+1}$ for $x>0$ is convex or concave. Answer:
• The function $f$ is convex on .
• The function $f$ is concave on .
Open intervals can be entered in the form $\left(a;b\right)$, closed intervals in the form $\left[a;b\right]$. $a$ and $b$ can be arbitrary expressions. For entering an interval, do not use the notation $\right]a;b\left[$ for open intervals. In your answer, enter infty for $\infty$.

##### Exercise 7.4.7
Consider the function $f:\left[-4.5;4\right]\to ℝ$ with $f\left(0\right):=2$. Its derivative $f\text{'}$ has the graph shown in the figure below:

1. Where is the function $f$ monotonically increasing and where it is monotonically decreasing? Find the maximum open intervals $\text{}\right]c;d\left[\text{}$ on which $f$ has this property.
2. What can you say about the maximum and minimum points of the function $f$?

• The function $f$ is monotonically
on $\text{}\right]-4.5;$
$\left[\text{}$.
• The function $f$ is monotonically
on $\text{}\right]$
$;0\left[\text{}$.
• The function $f$ is monotonically
• The function $f$ is monotonically
on $\text{}\right]3;4\left[\text{}$.
The maximum point of $f$ is at
. The minimum point of $f$ is at .