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

*Exercises*

Give an example of a continuous function

which takes exactly two values.

Let

be a continuous function which takes only finitely many values. Prove that is constant.

Does there exist a real number such that its third power, reduced by the fourfold of its second power, equals the square root of ?

Find a zero for the function

in the interval using the interval bisection method with a maximum error of .

We consider the function

Determine, starting with the interval and using the bisection method, an interval of length which contains a zero of .

We consider the function

Determine, starting with the interval and using the bisection method, an interval of length which contains a zero of .

We consider the mapping given by

Show, using the intermediate value theorem, that obtains every value at least in two points.

Let be a continuous function and let be "close“ to a zero of . Is then close to ?

Fridolin says:

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

we have and . Due to the Intermediate value theorem, there must be a zero between and , hence a number with . However, we always have .“

Where is the mistake in this argument?

Let be a real number. Show that the following properties are equivalent.

- There exist a polynomial , , with integer coefficients and with .
- There exists a polynomial , , wit .
- There exists a normed polynomial with .

Let

be continuous functions with and . Show that there is a point with .

The next exercises use following terms.

Let be a set and let

be a mapping. An element
such that
is called a
*fixed point*

Determine the fixed points of the mapping

Let be a polynomial of degree , . Show that has at most fixed points.

Let be a continuous function, and suppose that there exist with

and

Show that has a fixed point.

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

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

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

Let be a real interval and let

denote a continuous injective function. Show that is strictly increasing or strictly decreasing.

Show that the function defined by

is a continuous, strictly increasing, bijective function

and that its inverse function is also continuous.

- Sketch the graphs of the functions
and

- Determine the intersection points of these graphs.

Show that for every real number , there exists a continuous function

such that is the only zero of .

Show that for every real number , there exists a continuous function

such that is the only zero of and such that for every rational number , also is rational.

Show that for every real number , there exists a strictly increasing continuous function

such that is the only zero of and such that for every rational number , also is rational.

Let

be a continuous function. Show that is not surjective.

Give an example of a bounded interval and a continuous function

Let

be a continuous function defined over a real interval. The function has at points , , local maxima. Prove that the function has between and has at least one local minimum.

Determine directly, for which the power function

*Hand-in-exercises*

### Exercise (5 marks)

Find for the function

a zero in the interval using the interval bisection method, with a maximum error of .

### Exercise (3 marks)

Let denote a continuous function having the property that the image of is unbounded in both directions. Show that 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 of degree computes a zero within an accuracy of a given number 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

with
and
(hence, the coefficients of the polynomial, the accuracy and 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 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

be a continuous function from the interval into itself. Prove that has a fixed point.

### Exercise (2 marks)

Determine the limit of the sequence

### Exercise (2 marks)

Determine the minimum of the function

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