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

Exercises

### Exercise

Prove that the function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x\vert {x}\vert ,}$

is differentiable but not twice differentiable.

### Exercise

Let ${\displaystyle {}P\in \mathbb {R} [X]}$ be a polynomial, ${\displaystyle {}a\in \mathbb {R} }$ and ${\displaystyle {}n\in \mathbb {N} }$. Prove that ${\displaystyle {}P}$ is a multiple of ${\displaystyle {}(X-a)^{n}}$ if and only if ${\displaystyle {}a}$ is a zero of all the derivatives ${\displaystyle {}P,P^{\prime },P^{\prime \prime },\ldots ,P^{(n-1)}}$.

### Exercise

Consider the function

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

defined by

${\displaystyle {}f(x)={\begin{cases}x-\lfloor x\rfloor ,{\text{ if }}\lfloor x\rfloor {\text{ is even}},\\\lfloor x\rfloor -x+1,{\text{ if }}\lfloor x\rfloor {\text{ is odd}}\,.\end{cases}}\,}$

Examine ${\displaystyle {}f}$ in terms of continuity, differentiability and extremes.

### Exercise

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 ${\displaystyle {}42}$?

### Exercise

Determine local and global extrema of the function

${\displaystyle f\colon [-2,5]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=2x^{3}-5x^{2}+4x-1.}$

### Exercise

Determine local and global extrema of the function

${\displaystyle f\colon [-4,4]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=3x^{3}-7x^{2}+6x-3.}$

### Exercise

Consider the function

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

Find the point ${\displaystyle {}a\in [-3,3]}$ such that the tangent of the function at ${\displaystyle {}a}$ is parallel to the secant between ${\displaystyle {}-3}$ and ${\displaystyle {}3}$.

### Exercise

The city ${\displaystyle {}S=(0,0)}$ shall be connected by rails with the two cities ${\displaystyle {}T=(a,b)}$ and ${\displaystyle {}U=(a,-b)}$ with ${\displaystyle {}a\geq 0}$, ${\displaystyle {}b>0}$. The rails shall run along the ${\displaystyle {}x}$-axis until it ramifies into the two directions. Determine the ramification point, with as few rails as possible.

### Exercise

Next to a rectilinear river we want to fence a rectangular area of ${\displaystyle {}1000m^{2}}$, 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?

### Exercise

We consider the function

${\displaystyle {}F(x)=x^{4}-x^{3}\,.}$
1. Determine the zeroes of this function.
2. Determine on which intervals the function is positive or negative is.
3. Determine the extrema of this function.

### Exercise

Let

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

be differentiable functions. Let ${\displaystyle {}a\in \mathbb {R} }$ be a point, and suppose that

${\displaystyle f(a)=g(a){\text{ and }}f'(x)=g'(x){\text{ for all }}x.}$

Show that

${\displaystyle f(x)=g(x){\text{ for all }}x{\text{ gilt}}.}$

### Exercise

Let

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

be two differentiable functions. Let ${\displaystyle {}a\in \mathbb {R} }$. Suppose we have that

${\displaystyle f(a)\geq g(a){\text{ and }}f'(x)\geq g'(x){\text{ for all }}x\geq a.}$

Prove that

${\displaystyle f(x)\geq g(x){\text{ for all }}x\geq a.}$

### Exercise

Let

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

be a continuously differentiable function, and suppose that its graph intersects the diagonal in at least two points ${\displaystyle {}P\neq Q}$. Show that the graph of the derivative ${\displaystyle {}f'}$ has an intersection point with the line given by ${\displaystyle {}y=1}$.

### Exercise

Prove that a real polynomial function

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

of degree ${\displaystyle {}d\geq 1}$ has at most ${\displaystyle {}d-1}$ extrema, and moreover the real numbers can be divided into at most ${\displaystyle {}d}$ sections, where ${\displaystyle {}f}$ is strictly increasing or strictly decreasing.

### Exercise

Let ${\displaystyle {}I}$ be a real interval,

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

a twice continuously differentiable function, and let ${\displaystyle {}a\in I}$ be an inner point of the interval. Suppose that ${\displaystyle {}f'(a)=0}$. Show the following statements.

1. If ${\displaystyle {}f^{\prime \prime }(a)>0}$ holds, then ${\displaystyle {}f}$ has an isolated local minimum in ${\displaystyle {}a}$.
2. If ${\displaystyle {}f^{\prime \prime }(a)<0}$ holds, then ${\displaystyle {}f}$ has an isolated local maximum in ${\displaystyle {}a}$.

### Exercise

Let ${\displaystyle {}D\subseteq \mathbb {R} }$ and

${\displaystyle F\colon D\longrightarrow \mathbb {R} }$

be a rational function. Prove that ${\displaystyle {}F}$ is a polynomial if and only if there is a higher derivative such that ${\displaystyle {}F^{(n)}=0}$.

### Exercise

Let ${\displaystyle {}f\colon \mathbb {R} \rightarrow \mathbb {R} }$ be an ${\displaystyle {}n}$-fold continuously differentiable function with the property that its ${\displaystyle {}n}$-th derivative is everywhere positive. Show that ${\displaystyle {}f}$ has at most ${\displaystyle {}n}$ zeroes.

### Exercise

Discuss the following properties of the rational function

${\displaystyle f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {2x-3}{5x^{2}-3x+4}},}$

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

### Exercise

Consider

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

a) Prove that the function ${\displaystyle {}f}$ has in the real interval ${\displaystyle {}[0,1]}$ exactly one zero.

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

c) Find a rational number ${\displaystyle {}q\in [0,1]}$ such that ${\displaystyle {}\vert {f(q)}\vert \leq {\frac {1}{10}}}$.

### Exercise

Show that the function

${\displaystyle f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto e^{-{\frac {1}{x}}}\cdot \ln x}$

is bounded from below.

### Exercise

Let ${\displaystyle {}f\colon I\rightarrow \mathbb {R} }$ be a continuously differentiable function (defined on an open interval), and let ${\displaystyle {}a\in I}$ be a point with ${\displaystyle {}f'(a)\neq 0}$. Show that there exist open intervals ${\displaystyle {}J\subseteq I}$ with ${\displaystyle {}a\in J}$ and ${\displaystyle {}J'\subseteq \mathbb {R} }$, such that the restricted function ${\displaystyle {}f\colon J\rightarrow J'}$ is bijective.

### Exercise

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

### Exercise

Determine the limit of

${\displaystyle {\frac {x^{2}-3x+2}{x^{3}-2x+1}}}$

at the point ${\displaystyle {}x=1}$, and specifically

a) by polynomial division.

b) by the rule of l'Hospital.

### Exercise

Determine the limit

${\displaystyle \operatorname {lim} _{x\rightarrow 2}\,{\frac {3x^{2}-5x-2}{x^{3}-4x^{2}+x+6}}}$

by polynomial long division.

### Exercise

Determine the limit of the rational function

${\displaystyle {\frac {x^{3}-2x^{2}+x+4}{x^{2}+x}}}$

at the point ${\displaystyle {}a=-1}$.

### Exercise

Determine the limit

${\displaystyle \operatorname {lim} _{x\rightarrow 1}\,{\frac {\sqrt {1-x}}{\sqrt[{3}]{1-x^{2}}}}.}$

Hand-in-exercises

### Exercise (5 marks)

From a sheet of paper with side lengths of ${\displaystyle {}20}$ cm and ${\displaystyle {}30}$ 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

${\displaystyle f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {3x^{2}-2x+1}{x-4}},}$

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

${\displaystyle {}f(x)={\frac {ax+b}{cx+d}}\,}$

(with ${\displaystyle {}a,b,c,d\in \mathbb {R} }$, ${\displaystyle {}a,c\neq 0}$,) has no local extrema.

### Exercise (4 marks)

Let

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

be a polynomial function of degree ${\displaystyle {}d\geq 1}$. Let ${\displaystyle {}m}$ be the number of local maxima of ${\displaystyle {}f}$ and ${\displaystyle {}n}$ the number of local minima of ${\displaystyle {}f}$. Prove that if ${\displaystyle {}d}$ is odd then ${\displaystyle {}m=n}$ and that if ${\displaystyle {}d}$ is even then

${\displaystyle {}\vert {m-n}\vert =1\,.}$

### Exercise (3 marks)

Determine the limit of the rational function

${\displaystyle {\frac {x^{4}+2x^{3}-3x^{2}-4x+4}{2x^{3}-x^{2}-4x+3}}}$

at the point ${\displaystyle {}a=1}$.