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

    Exercises

### Exercise

Prove that the function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x\vert {x}\vert ,$
is differentiable but not twice differentiable.

### Exercise

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

### Exercise

Consider the function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,$
defined by
    $f(x)={\begin{cases}x-\lfloor x\rfloor ,{\text{ if }}\lfloor x\rfloor {\text{ is even}},\\\lfloor x\rfloor -x+1,{\text{ if }}\lfloor x\rfloor {\text{ is odd}}\,.\end{cases}}\,$
Examine $f$ 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 $42$?

### Exercise

Determine local and global extrema of the function
    $f\colon [-2,5]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=2x^{3}-5x^{2}+4x-1.$

### Exercise

Determine local and global extrema of the function
    $f\colon [-4,4]\longrightarrow \mathbb {R} ,x\longmapsto f(x)=3x^{3}-7x^{2}+6x-3.$

### Exercise

Consider the function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x)=4x^{3}+3x^{2}-x+2.$
Find the point $a\in [-3,3]$ such that the tangent of the function at $a$ is parallel to the secant between $-3$ and $3$.

### Exercise

The city $S=(0,0)$ shall be connected by rails with the two cities $T=(a,b)$ and $U=(a,-b)$ with $a\geq 0$, $b>0$. The rails shall run along the $x$-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 $1000m^{2}$, 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
    $F(x)=x^{4}-x^{3}\,.$
* Determine the zeroes of this function.
* Determine on which intervals the function is positive or negative is.
* Determine the extrema of this function.

### Exercise

Let
    $f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$
be differentiable functions. Let $a\in \mathbb {R}$ be a point, and suppose that
    $f(a)=g(a){\text{ and }}f'(x)=g'(x){\text{ for all }}x.$
Show that
    $f(x)=g(x){\text{ for all }}x{\text{ gilt}}.$

### Exercise

Let
    $f,g\colon \mathbb {R} \longrightarrow \mathbb {R}$
be two differentiable functions. Let $a\in \mathbb {R}$. Suppose we have that
    $f(a)\geq g(a){\text{ and }}f'(x)\geq g'(x){\text{ for all }}x\geq a.$
Prove that
    $f(x)\geq g(x){\text{ for all }}x\geq a.$

### Exercise

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

### Exercise

Prove that a real polynomial function
    $f\colon \mathbb {R} \longrightarrow \mathbb {R}$
of degree $d\geq 1$ has at most $d-1$ extrema, and moreover the real numbers can be divided into at most $d$ sections, where $f$ is strictly increasing or strictly decreasing.

### Exercise

Let $I$ be a real interval,
    $f\colon I\longrightarrow \mathbb {R}$
a twice continuously differentiable function, and let $a\in I$ be an inner point of the interval. Suppose that $f'(a)=0$. Show the following statements.
* If $f^{\prime \prime }(a)>0$ holds, then $f$ has an isolated local minimum in $a$.
* If $f^{\prime \prime }(a)<0$ holds, then $f$ has an isolated local maximum in $a$.

### Exercise

Let $D\subseteq \mathbb {R}$ and
    $F\colon D\longrightarrow \mathbb {R}$
be a rational function. Prove that $F$ is a polynomial if and only if there is a higher derivative such that $F^{(n)}=0$.

### Exercise

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

### Exercise

Discuss the following properties of the rational function
    $f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {2x-3}{5x^{2}-3x+4}},$
domain, zeros, growth behavior, (local) extrema. Sketch the graph of the function.

### Exercise

Consider
    $f(x)=x^{3}+x-1.$
a) Prove that the function $f$ has in the real interval $[0,1]$ exactly one zero.
b) Compute the first decimal digit in the decimal system of this zero point.
c) Find a rational number $q\in [0,1]$ such that $\vert {f(q)}\vert \leq {\frac {1}{10}}$.

### Exercise

Show that the function
    $f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto e^{-{\frac {1}{x}}}\cdot \ln x$
is bounded from below.

### Exercise

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

### Exercise

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

### Exercise

Determine the limit of
    ${\frac {x^{2}-3x+2}{x^{3}-2x+1}}$
at the point $x=1$, and specifically
a) by polynomial division.
b) by the rule of l'Hospital.

### Exercise

Determine the limit
    $\operatorname {lim} _{x\rightarrow 2}\,{\frac {3x^{2}-5x-2}{x^{3}-4x^{2}+x+6}}$
by polynomial long division.

### Exercise

Determine the limit of the rational function
    ${\frac {x^{3}-2x^{2}+x+4}{x^{2}+x}}$
at the point $a=-1$.

### Exercise

Determine the limit
    $\operatorname {lim} _{x\rightarrow 1}\,{\frac {\sqrt {1-x}}{\sqrt[{3}]{1-x^{2}}}}.$
    Hand-in-exercises

### Exercise (5 marks)

From a sheet of paper with side lengths of $20$ cm and $30$ 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
    $f\colon D\longrightarrow \mathbb {R} ,x\longmapsto f(x)={\frac {3x^{2}-2x+1}{x-4}},$
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
    $f(x)={\frac {ax+b}{cx+d}}\,$
(with $a,b,c,d\in \mathbb {R}$, $a,c\neq 0$,) has no local extrema.

### Exercise (4 marks)

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

### Exercise (3 marks)

Determine the limit of the rational function
    ${\frac {x^{4}+2x^{3}-3x^{2}-4x+4}{2x^{3}-x^{2}-4x+3}}$
at the point $a=1$.