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Elements of Real Analysis

Robert G. Bartle

Chapter 21

Mapping Theorems and Extremum Problems - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $f$ be the mapping of $\mathbf{R}^2$ into $\mathbf{R}^2$ which sends the point $(x, y)$ into the point $(u, v)$ given by
$$
u=x+y, \quad v=2 x+a y .
$$

Calculate the derivative $D f$. Show that $D f$ is one-one if and only if it maps $\mathbf{R}^2$ onto $\mathbf{R}^2$, and that this is the case if and only if $a \neq 2$. Examine the image of the unit square $\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\}$ in the three cases $a=1, a=2$, $a=3$.

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Problem 2

Let $f$ be the mapping of $\mathbf{R}^2$ into $\mathbf{R}^2$ which sends the point $(x, y)$ into the point $(u, v)$ given by
$$
u=x, \quad v=x y .
$$

Draw some curves $u=$ constant, $v=$ constant in the $(x, y)$-plane and some curves $x=$ constant, $y=$ constant in the $(u, v)$-plane. Is this mapping one-one? Does $f$ map onto all of $\mathbf{R}^2$ ? Show that if $x \neq 0$, then $f$ maps some neighborhood of $(x, y)$ in a one-one fashion onto a neighborhood of $(x, x y)$. Into what region in the $(u, v)$-plane does $f$ map the rectangle $\{(x, y): 1 \leq x \leq 2,0 \leq y \leq 2\}$ ? What points in the $(x, y)$-plane map under $f$ into the rectangle $\{(u, v): 1 \leq u \leq 2$, $0 \leq v \leq 2]$ ?

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03:43

Problem 3

Let $f$ be the mapping of $\mathbf{R}^2$ into $\mathbf{R}^2$ which sends the point $(x, y)$ into the point $(u, v)$ given by
$$
u=x^2-y^2, \quad \forall=2 x y .
$$

What curves in the $(x, y)$-plane map under $f$ into the lines $u=$ constant, $v=$ constant? Into what curves in the $(u, v)$-plane do the lines $x=$ constant, $y=$ constant map? Show that each non-zero point $(u, v)$ is the image under $f$ of two points. Into what region does $f$ map the square $\{(x, y): 0 \leq x \leq 1$, $0 \leq y \leq 1\}$ ? What region is mapped by $f$ into the square $\{(u, v): 0 \leq u \leq 1$, $0 \leq v \leq 1\}$ ?

Hossam Mohamed
Hossam Mohamed
Numerade Educator

Problem 4

Let $f$ be the mapping in the preceding exercise. Show that $f$ is locally one-one at every point except $\theta=(0,0)$, but $f$ is not one-one on $\mathrm{R}^2$.

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06:36

Problem 5

Let $f$ be in Class $C^{\prime}$ on $\mathbf{R}^p$ onto $\mathbf{R}^{\circ}$ and suppose that $f$ has an inverse. Is it true that for each $x$ in $\mathrm{R}^p$, then $D f(x)$ is a one-one linear function which maps $\mathrm{R}^p$ onto $\mathrm{R}^0$ ?

Anurag Kumar
Anurag Kumar
Numerade Educator
01:08

Problem 6

Let $f$ be defined on $\mathbf{R}$ to $\mathbf{R}$ by
$$
\begin{aligned}
f(x) & =x+2 x^2 \sin (1 / x), & & x \neq 0, \\
& =0, & & x=0 .
\end{aligned}
$$

Then $D f(0)$ is one-one but $f$ has no inverse near $x=0$.

Aman Gupta
Aman Gupta
Numerade Educator

Problem 7

Let $f$ be a function on $\mathrm{R}^p$ to $\mathrm{R}^p$ which is differentiable on a neighborhood of a point $c$ and such that $D f(c)$ has an inverse. Then is it true that $f$ has an inverse on a neighborhood of $c$ ?

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24:19

Problem 8

Let $f$ be a function on $\mathbf{R}^p$ to $\mathrm{R}^p$. If $f$ is differentiable at $c$ and has a differentiable inverse, then is it true that $D f(c)$ is one-one?

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator

Problem 9

Suppose that $f$ is differentiable on a neighborhood of a point $c$ and that if $\epsilon>0$ then there exists $\delta(\epsilon)>0$ such that if $|x-c|<\delta(\epsilon)$, then $\mid D f(x)(z)-$ $D f(c)(z)|\leq \mathrm{e}| z \mid$ for all $z$ in $\mathrm{R}^p$. Prove that the partial derivatives of $f$ exist and are continuous at $c$.

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Problem 10

Suppose that $L_0$ is a one-one linear function on $R^p$ to $R^q$. Show that there exists a positive number $\alpha$ such that if $L$ is a linear function on $\mathbf{R}^p$ to $\mathbf{R}^p$ satisfying
$$
\left|L(z)-L_0(z)\right| \leq \alpha|z| \quad \text { for } \quad z \in \mathbf{R}^p,
$$
then $L$ is one-one.

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Problem 11

Suppose that $L_0$ is a linear function on $\mathbf{R}^p$ with range all of $\mathbf{R}^{\bullet}$. Show that there exists a positive number $\beta$ such that if $L$ is a linear function on $R^p$ into $R^{\varepsilon}$ satisfying
$$
\left|L(z)-L_0(z)\right| \leq \beta|z| \quad \text { for } \quad z \in \mathrm{R}^p,
$$
then the range of $L$ is $\mathbf{R} q$.

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01:36

Problem 12

Let $f$ be in Class $C^{\prime}$ on a neighborhood of a point $c$ in $\mathrm{R}^p$ and with values in $R^p$. If $D f(c)$ is one-one and has range equal to $R^p$, then there exists a positive number $\delta$ such that if $|x-c|<\delta$, then $D f(x)$ is one-one and has range equal to $\mathbf{R}^p$.

Stanley Enemuo
Stanley Enemuo
Numerade Educator

Problem 13

Let $f$ be defined on $\mathbf{R}^2$ to $\mathbf{R}^2$ by $f(x, y)=(x \cos y, x \sin y)$. Show that if $x_0>0$, then there exists a neighborhood of $\left(x_0, y_0\right)$ on which $f$ is one-one, but that there are infinitely many points which are mapped into $f\left(x_0, y_0\right)$.

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Problem 14

Let $F$ be defined on $\mathbf{R} \times \mathbf{R}$ to $\mathbf{R}$ by $F(x, y)=x^2-y$. Show that $F$ is in Class $C^{\prime}$ on a neighborhood of $(0,0)$ but there does not exist a continuous function $\varphi$ defined on a neighborhood of 0 such that $F[\varphi(y), y]=0$.

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Problem 15

Suppose that, in addition to the hypotheses of the Implicit Function Theorem 21.11, the function $F$ has continuous partial derivatives of order $n$. Show that the solution function $\varphi$ has continuous partial derivatives of order $n$.

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02:29

Problem 16

Let $F$ be the function on $\mathbf{R}^2 \times \mathbf{R}^2$ to $\mathbf{R}^2$ defined for $x=\left(\xi_1, \xi_2\right)$ and $y=\left(\eta_1, \eta_2\right)$ by the formula
$$
F(x, y)=\left(\xi_1{ }^3+\xi_2 \eta_1+\eta_2, \xi_1 \eta_2+\xi_2{ }^3-\eta_1\right) .
$$

At what points $(x, y)$ can one solve the equation $F(x, y)=\theta$ for $x$ in terms of $y$. Calculate the derivative of this solution function, when it exists. In particular, calculate the partial derivatives of the coordinate functions of $\varphi$ with respect to $\eta_1, \eta_2$.

Aman Gupta
Aman Gupta
Numerade Educator

Problem 17

Let $f$ be defined and continuous on the set $\mathbb{D}=\left\{x \in \mathrm{R}^p:|x| \leq 1\right\}$ with values in $\mathrm{R}$. Suppose that $f$ is differentiable at every interior point of $\mathbb{D}$ and that $f(x)=0$ for all $|x|=1$. Prove that there exists an interior point $c$ of $D$ and that $D f(c)=0$ (This result may be regarded as a generalization of Rolle's Theorem.)

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01:24

Problem 18

If we define $f$ on $\mathrm{R}^2$ to $\mathrm{R}$ by
$$
f(\xi, \eta)=\xi^2+4 \xi \eta+\eta^2,
$$
then the origin is not a relative extreme point but a saddle point of $f$.

Lucas Finney
Lucas Finney
Numerade Educator
07:38

Problem 19

(a) Let $f_1$ be defined on $\mathrm{R}^2$ to $\mathrm{R}$ by
$$
f_1(\xi, \eta)=\xi^4+\eta^4,
$$
then the origin $\theta=(0,0)$ is a relative minimum of $f_1$ and $\Delta=0$ at $\theta$. (Here $\left.\Delta=f_{k t} f_{\eta \eta}-f_{k \eta}{ }^2+\right)$
(b) If $f_2=-f_1$, then the origin is a relative maximum of $f_2$ and $\Delta=0$ at $\theta$.
(c) If $f_3$ is defined on $\mathbf{R}^2$ to $\mathbf{R}$ by
$$
f_3(\xi, \eta)=\xi^4-\eta^4,
$$
then the origin $\theta=(0,0)$ is a saddle point of $f_3$ and $\Delta=0$ at $\theta$. (The moral of this exercise is that if $\Delta=0$, then anything can happen.)

Muhammad Saleem
Muhammad Saleem
Numerade Educator
02:04

Problem 20

Let $f$ be defined on $D=\left\{(\xi, \eta) \in \mathbf{R}^2: \xi>0, \eta>0\right\}$ to $\mathbf{R}$ by the formula
$$
f(\xi, \eta)=\frac{1}{\xi}+\frac{1}{\eta}+c \xi \eta
$$

Locate the critical points of $f$ and determine whether they yield relative maxima, relative minima, or saddle points. If $c>0$ and we set
$$
D_1=\{(\xi, \eta): \xi>0, \eta>0, \xi+\eta \leq c\},
$$
then locate the relative extrema of $f$ on $\mathfrak{D}_1$.

Lucas Finney
Lucas Finney
Numerade Educator
08:04

Problem 21

Suppose we are given $n$ points $\left(\xi_i, \eta_j\right)$ in $\mathbf{R}^2$ and desire to find the linear function $F(x)=A x+B$ for which the quantity
$$
\sum_{j=1}^n\left[F\left(\xi_j\right)-\eta_j\right]^2
$$
is minimized. Show that this leads to the equations
$$
\begin{aligned}
A \sum_{j=1}^n \xi_j{ }^2+B \sum_{j=1}^n \xi_j & =\sum_{j=1}^n \xi_j \eta_j, \\
A \sum_{j=1}^n \xi_j+n B & =\sum_{j=1}^n \eta_j,
\end{aligned}
$$
for the numbers $A, B$. This linear function is referred to as the linear function which best fits the given $n$ points in the sense of least squares.

Stella Li
Stella Li
Numerade Educator

Problem 22

Let $f$ be defined and continuous on the set $D=\left\{x \in \mathbf{R}^p:|x| \leq 1\right\}$ with values in $\mathbf{R}$. If $f$ is differentiable at every interior point of $\Phi$ and if
$$
\sum_{j=1}^p f_{i j k}(x)=0
$$
for all $|x|<1$, then $f$ is said to be harmonic in D. Suppose that $f$ is not constan $t$ and that $f$ does not attain its supremum on $C=\{x:|x|=1\}$ but at a point $c$ interior to $D$. Then, if $\epsilon>0$ is sufficiently small, the function $g$ defined by
$$
g(x)=f(x)+\epsilon x-\left.c\right|^2
$$
does not attain its supremum on $C$ but at some interior point $c^{\prime}$. Since
$$
g_{i j j}\left(c^{\prime}\right)=f_{t j k j}\left(c^{\prime}\right)+2 \mathrm{e}, \quad j=1, \ldots, p,
$$
it follows that
$$
\sum_{j=1}^p g_{k ; i j}\left(c^{\prime}\right)=2 \varepsilon p>0,
$$
so that some $g_{\mathrm{fj} f\left(c^{\prime}\right)}>0$, a contradiction. (Why?) Therefore, if $f$ is harmonic in $D$ it attains its supremum (and also its infimum) on $C$. Show also that if $f$ and $h$ are harmonic in $D$ and $f(x)=h(x)$ for $x \in C$, then $f(x)=h(x)$ for $x \in \mathbb{D}$.

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03:31

Problem 23

Show that the function
$$
f(\xi, \eta)=\left(\eta-\xi^2\right)\left(\eta-2 \xi^2\right)
$$
does not have a relative extremum at $\theta=(0,0)$ although it has a relative minimum along every line $\xi=\alpha t, \eta=\beta t$.

Aman Gupta
Aman Gupta
Numerade Educator
05:25

Problem 24

Find the dimensions of the box of maximum volume which can be fitted into the ellipsoid
$$
\frac{\xi^2}{a^2}+\frac{\eta^2}{b^2}+\frac{\zeta^2}{c^2}=1
$$
assuming that each edge of the box is parallel to a coordinate axis.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
07:15

Problem 25

(a) Find the maximum of
$$
f\left(x_1, x_2, \ldots, x_n\right)=\left(x_1 x_2 \cdots x_n\right)^2,
$$
subject to the constraint
$$
x_1^2+x_2^2+\cdots+x_n^2=1 .
$$
(b) Show that the geometric mean of a collection of non-negative real numbers $\left\{a_1, a_2, \ldots, a_n\right\}$ does not exceed their arithmetic mean; that is,
$$
\left(a_1 a_2 \cdots a_n\right)^{1 / n} \leq \frac{1}{n}\left(a_1+a_2+\cdots+a_n\right) .
$$

WZ
Wen Zheng
Numerade Educator

Problem 26

(a) Let $p>1, q>1$, and $\frac{1}{p}+\frac{1}{q}=1$. Show that the minimum of
$$
f(\xi, \eta)=\frac{\xi^p}{p}+\frac{\eta^p}{q},
$$
subject to the constraint $\xi \eta=1$, is 1 .
(b) From (a), show that if $a, b$ are non-negative real numbers, then
$$
a b \leq \frac{a^p}{p}+\frac{b^e}{q} .
$$
(c) Let $\left\{a_j\right\},\left\{b_j\right\}, j=1, \ldots, n$, be non-negative real numbers, and obtain Hölder's Inequality:
$$
\sum_{j=1}^n a_j b_j \leq\left(\sum_{j=1}^n a_j\right)^{1 / p}\left(\sum_{j=1}^n b_j\right)^{1 / q} .
$$
(d) Note that
$$
|a+b|^p=|a+b||a+b|^{p / q} \leq|a||a+b|^{[p / q}+|b||a+b|^{p / q} .
$$
Use Hölder's Inequality in (c) and derive the Minkowski Inequality
$$
\left(\sum_{j=1}^n\left|a_i+b_j\right|^p\right)^{1 / p} \leq\left(\sum_{j=1}^n\left|a_j\right|^p\right)^{1 / p}+\left(\sum_{j=1}^n\left|b_j\right|^p\right)^{1 / p} .
$$

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