# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 11

Exercises

### Exercise

Give an example of a continuous function

${\displaystyle f\colon \mathbb {Q} \longrightarrow \mathbb {R} ,}$

which takes exactly two values​​.

### Exercise

Let

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$

be a continuous function which takes only finitely many values. Prove that ${\displaystyle {}f}$ is constant.

### Exercise

Does there exist a real number such that its third power, reduced by the fourfold of its second power, equals the square root of ${\displaystyle {}42}$?

### Exercise

Find a zero for the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{2}+x-1,}$

in the interval ${\displaystyle {}[0,1]}$ using the interval bisection method with a maximum error of ${\displaystyle {}1/100}$.

### Exercise

We consider the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{3}+4x^{2}-x+3.}$

Determine, starting with the interval ${\displaystyle {}[-5,-4]}$ and using the bisection method, an interval of length ${\displaystyle {}1/8}$ which contains a zero of ${\displaystyle {}f}$.

### Exercise

We consider the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{3}-4x+2.}$

Determine, starting with the interval ${\displaystyle {}[1,2]}$ and using the bisection method, an interval of length ${\displaystyle {}1/8}$ which contains a zero of ${\displaystyle {}f}$.

### Exercise

We consider the mapping ${\displaystyle {}f\colon \mathbb {R} \setminus {\{0,1\}}\rightarrow \mathbb {R} }$ given by

${\displaystyle {}f(x)={\frac {1}{x^{3}}}+{\frac {1}{(x-1)^{3}}}\,.}$

Show, using the intermediate value theorem, that ${\displaystyle {}f}$ obtains every value ${\displaystyle {}c\neq 0}$ at least in two points.

### Exercise

Let ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ be a continuous function and let ${\displaystyle {}x}$ be "close“ to a zero of ${\displaystyle {}f}$. Is then ${\displaystyle {}f(x)}$ close to ${\displaystyle {}0}$?

### Exercise

Fridolin says:

"Something is wrong about the Intermediate value theorem. For the continuous function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},}$

we have ${\displaystyle {}f(-1)=-1}$ and ${\displaystyle {}f(1)=1}$. Due to the Intermediate value theorem, there must be a zero between ${\displaystyle {}-1}$ and ${\displaystyle {}1}$, hence a number ${\displaystyle {}x\in [-1,1]}$ with ${\displaystyle {}f(x)=0}$. However, we always have ${\displaystyle {}{\frac {1}{x}}\neq 0}$.“

Where is the mistake in this argument?

### Exercise

Let ${\displaystyle {}z\in \mathbb {R} }$ be a real number. Show that the following properties are equivalent.

1. There exist a polynomial ${\displaystyle {}P\in \mathbb {R} [X]}$, ${\displaystyle {}P\neq 0}$, with integer coefficients and with ${\displaystyle {}P(z)=0}$.
2. There exists a polynomial ${\displaystyle {}Q\in \mathbb {Q} [X]}$, ${\displaystyle {}Q\neq 0}$, wit ${\displaystyle {}Q(z)=0}$.
3. There exists a normed polynomial ${\displaystyle {}R\in \mathbb {Q} [X]}$ with ${\displaystyle {}R(z)=0}$.

### Exercise

Let

${\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }$

be continuous functions with ${\displaystyle {}f(a)\geq g(a)}$ and ${\displaystyle {}f(b)\leq g(b)}$. Show that there is a point ${\displaystyle {}c\in [a,b]}$ with ${\displaystyle {}f(c)=g(c)}$.

The next exercises use following terms.

Let ${\displaystyle {}M}$ be a set and let

${\displaystyle f\colon M\longrightarrow M}$

be a mapping. An element ${\displaystyle {}x\in M}$ such that ${\displaystyle {}f(x)=x}$ is called a fixed point

of ${\displaystyle {}f}$.

### Exercise

Determine the fixed points of the mapping

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}.}$

### Exercise

Let ${\displaystyle {}P\in \mathbb {R} [X]}$ be a polynomial of degree ${\displaystyle {}d\geq 1}$, ${\displaystyle {}P\neq X}$. Show that ${\displaystyle {}P}$ has at most ${\displaystyle {}d}$ fixed points.

### Exercise

Let ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ be a continuous function, and suppose that there exist ${\displaystyle {}x,y\in \mathbb {R} }$ with

${\displaystyle {}f(x)\leq x\,}$

and

${\displaystyle {}f(y)\geq y\,.}$

Show that ${\displaystyle {}f}$ has a fixed point.

### Exercise

Show that the image of a closed interval under a continuous function is not necessarily closed.

### Exercise

Show that the image of an open interval under a continuous function is not necessarily open.

### Exercise

Show that the image of a bounded interval under a continuous function is not necessarily bounded.

### Exercise

Let ${\displaystyle {}I}$ be a real interval and let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

denote a continuous injective function. Show that ${\displaystyle {}f}$ is strictly increasing or strictly decreasing.

### Exercise

Show that the function defined by

${\displaystyle {}f(x)={\frac {x}{\vert {x}\vert +1}}\,}$

is a continuous, strictly increasing, bijective function

${\displaystyle f\colon \mathbb {R} \longrightarrow {]{-1},1[}}$

and that its inverse function is also continuous.

### Exercise

1. Sketch the graphs of the functions
${\displaystyle f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto x-1,}$

and

${\displaystyle g\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},}$
2. Determine the intersection points of these graphs.

### Exercise

Show that for every real number ${\displaystyle {}a\in \mathbb {R} }$, there exists a continuous function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$

such that ${\displaystyle {}a}$ is the only zero of ${\displaystyle {}f}$.

### Exercise

Show that for every real number ${\displaystyle {}x\in \mathbb {R} }$, there exists a continuous function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$

such that ${\displaystyle {}x}$ is the only zero of ${\displaystyle {}f}$ and such that for every rational number ${\displaystyle {}q}$, also ${\displaystyle {}f(q)}$ is rational.

### Exercise

Show that for every real number ${\displaystyle {}x\in \mathbb {R} }$, there exists a strictly increasing continuous function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$

such that ${\displaystyle {}x}$ is the only zero of ${\displaystyle {}f}$ and such that for every rational number ${\displaystyle {}q}$, also ${\displaystyle {}f(q)}$ is rational.

### Exercise

Let

${\displaystyle f\colon [0,1]\longrightarrow [0,1[}$

be a continuous function. Show that ${\displaystyle {}f}$ is not surjective.

### Exercise

Give an example of a bounded interval ${\displaystyle {}I\subseteq \mathbb {R} }$ and a continuous function

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$
such that the image of ${\displaystyle {}f}$ is bounded, but the function admits no maximum.

### Exercise

Let

${\displaystyle f\colon I\longrightarrow \mathbb {R} }$

be a continuous function defined over a real interval. The function has at points ${\displaystyle {}x_{1},x_{2}\in I}$, ${\displaystyle {}x_{1}, local maxima. Prove that the function has between ${\displaystyle {}x_{1}}$ and ${\displaystyle {}x_{2}}$ has at least one local minimum.

### Exercise

Determine directly, for which ${\displaystyle {}n\in \mathbb {N} }$ the power function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},}$
has an extremum at the point zero.

Hand-in-exercises

### Exercise (5 marks)

Find for the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=x^{3}-3x+1,}$

a zero in the interval ${\displaystyle {}[0,1]}$ using the interval bisection method, with a maximum error of ${\displaystyle {}1/200}$.

### Exercise (3 marks)

Let ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ denote a continuous function having the property that the image of ${\displaystyle {}f}$ is unbounded in both directions. Show that ${\displaystyle {}f}$ is surjective.

### Exercise (4 marks)

Show that a real polynomial of odd degree has at least one real zero.

### Exercise (5 marks)

Write a computer-program (in pseudocode) which for a polynomial ${\displaystyle {}dX^{3}+cX^{2}+bX+a}$ of degree ${\displaystyle {}3}$ computes a zero within an accuracy of a given number ${\displaystyle {}e>0}$ berechnet.

• The computer has as many memory units as needed, which can contain nonnegative real numbers.
• It can write the content of a memory unit into another memory unit.
• It can halve the content of a memory unit and write the result into another memory unit.
• It can add the content of two memory units and write the result into another memory unit.
• It can multiply the content of two memory units and write the result into another memory unit.
• It can compare the content of memory units and can, depending on the outcome, switch to a certain program line.
• It can print contents of memory units and it can print given texts.
• There is a stop command.

The initial configuration is

${\displaystyle (a,b,c,d,e,1,0,0,\ldots )}$

with ${\displaystyle {}a,b,c\geq 0}$ and ${\displaystyle {}d,e>0}$ (hence, the coefficients of the polynomial, the accuracy ${\displaystyle {}e}$ and ${\displaystyle {}1}$ are in the first memory units). The program shall print a sentence telling the bounds of an interval for a zero with the wished-for accuracy and stop.
Caution: The main difficulty is here that the polynomials do not have any zero on ${\displaystyle {}\mathbb {R} _{+}}$ due to our condition. Hence we have to find a zero in the negative real numbers. However, the memory units do not accept negative numbers. Therefore we have to emulate/simulate negative numbers by nonnegative numbers.

### Exercise (4 marks)

Let

${\displaystyle f\colon [a,b]\longrightarrow [a,b]}$

be a continuous function from the interval ${\displaystyle {}[a,b]}$ into itself. Prove that ${\displaystyle {}f}$ has a fixed point.

### Exercise (2 marks)

Determine the limit of the sequence

${\displaystyle x_{n}={\sqrt[{3}]{\frac {27n^{3}+13n^{2}+n}{8n^{3}-7n+10}}},\,n\in \mathbb {N} .}$

### Exercise (2 marks)

Determine the minimum of the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2}+3x-5.}$