#### 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 $f(x):=\frac{{x}^{2}-1}{{x}^{2}+1}$ is monotonically increasing or decreasing.

Answer:

Answer:

- $f$ is monotonically

on $\text{}]-\infty ;0[\text{}$.

- $f$ is monotonically

on $\text{}]0;\infty [\text{}$.

##### **Exercise 7.4.6 **

Specify the (maximum) open intervals $\text{}]c;d[\text{}$ on which the function $f$ with $f(x):=\frac{{x}^{2}-1}{{x}^{2}+1}$ for $x>0$ is convex or concave. Answer:

- The function $f$ is convex on .

- The function $f$ is concave on .

`infty`for $\infty $.##### **Exercise 7.4.7 **

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

Answer:

. The minimum point of $f$ is at .

- Where is the function $f$ monotonically increasing and where it is monotonically decreasing? Find the maximum open intervals $\text{}]c;d[\text{}$ on which $f$ has this property.

- What can you say about the maximum and minimum points of the function $f$?

Answer:

- The function $f$ is monotonically

on $\text{}]-4.5;$

$[\text{}$.

- The function $f$ is monotonically

on $\text{}]$

$;0[\text{}$.

- The function $f$ is monotonically

on .

- The function $f$ is monotonically

on $\text{}]3;4[\text{}$.

. The minimum point of $f$ is at .