# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 15

*Exercises*

Prove that the function

is differentiable but not twice differentiable.

Let be a polynomial, and . Prove that is a multiple of if and only if is a zero of all the derivatives .

Consider the function

defined by

Examine in terms of continuity, differentiability and extremes.

Does there exist a real number, which, in its fourth power, reduced by the double of its third power, equals the negative of the square root of ?

Determine local and global extrema of the function

Determine local and global extrema of the function

Consider the function

Find the point such that the tangent of the function at is parallel to the secant between and .

The city shall be connected by rails with the two cities and with , . The rails shall run along the -axis until it ramifies into the two directions. Determine the ramification point, with as few rails as possible.

Next to a rectilinear river we want to fence a rectangular area of , one side of the area is the river itself. For the other three sides, we need a fence. Which is the minimal length of the fence we need?

We consider the function

- Determine the zeroes of this function.
- Determine on which intervals the function is positive or negative is.
- Determine the extrema of this function.

Let

be differentiable functions. Let be a point, and suppose that

Show that

Let

be two differentiable functions. Let . Suppose we have that

Prove that

Let

be a continuously differentiable function, and suppose that its graph intersects the diagonal in at least two points . Show that the graph of the derivative has an intersection point with the line given by .

Prove that a real polynomial function

of degree has at most extrema, and moreover the real numbers can be divided into at most sections, where is strictly increasing or strictly decreasing.

Let be a real interval,

a twice continuously differentiable function, and let be an inner point of the interval. Suppose that . Show the following statements.

- If holds, then has an isolated local minimum in .
- If holds, then has an isolated local maximum in .

Let and

be a rational function. Prove that is a polynomial if and only if there is a higher derivative such that .

Let be an -fold continuously differentiable function with the property that its -th derivative is everywhere positive. Show that has at most zeroes.

Discuss the following properties of the rational function

domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.

Consider

a) Prove that the function has in the real interval exactly one zero.

b) Compute the first decimal digit in the decimal system of this zero point.

c) Find a rational number such that .

Show that the function

is bounded from below.

Let be a continuously differentiable function (defined on an open interval), and let be a point with . Show that there exist open intervals with and , such that the restricted function is bijective.

Prove the mean value theorem out of the second mean value theorem.

Determine the limit of

at the point , and specifically

a) by polynomial division.

b) by the rule of l'Hospital.

Determine the limit

by polynomial long division.

Determine the limit of the rational function

at the point .

Determine the limit

*Hand-in-exercises*

### Exercise (5 marks)

From a sheet of paper with side lengths of cm and cm we want to realize a box (without cover) with the greatest possible volume. We do it in this way. We remove from each corner a square of the same size, then we lift up the sides and we glue them. Which box height do we need to realize the maximum volume?

### Exercise (4 marks)

Discuss the following properties of the rational function

domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.

### Exercise (5 marks)

Prove that a non-constant rational function of the shape

(with , ,) has no local extrema.

### Exercise (4 marks)

Let

be a polynomial function of degree . Let be the number of local maxima of and the number of local minima of . Prove that if is odd then and that if is even then

### Exercise (3 marks)

Determine the limit of the rational function

at the point .

<< | Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >> PDF-version of this exercise sheet Lecture for this exercise sheet (PDF) |
---|