A Course in Calculus and Real Analysis

Sudhir R. Ghorpade, Balmohan V. Limaye

Chapter 4

Differentiation - all with Video Answers

Educators

+ 4 more educators

Chapter Questions

Problem 1

Use the definition of a derivative to find $f^{\prime}(x)$ if
(i) $f(x)=x^{2}, x \in \mathbb{R}$
(ii) $f(x)=1 / x, 0 \neq x \in \mathbb{R}$
(iii) $f(x)=\sqrt{x^{2}+1}, x \in \mathbb{R}$
(iv) $f(x)=1 / \sqrt{2 x+3}, x \in(-3 / 2, \infty)$.

Vikash Ranjan

Vikash Ranjan

Numerade Educator

Problem 2

Let $f:(a, b) \rightarrow \mathbb{R}$ be a function such that
$$
|f(x+h)-f(x)| \leq C|h|^{r} \quad \text { for all } x, x+h \in(a, b),
$$
where $C$ is a constant and $r \in \mathbb{Q}$ with $r \geq 1$. Show that $f$ is differentiable on $(a, b)$ and compute $f^{\prime}(x)$ for $x \in(a, b)$.

Apratim De

Numerade Educator

Problem 3

If $f:(a, b) \rightarrow \mathbb{R}$ is differentiable at $c \in(a, b)$, then show that
$$
\lim _{h \rightarrow 0^{+}} \frac{f(c+h)-f(c-h)}{2 h}
$$
exists and equals $f^{\prime}(c) .$ Is the converse true?

Sarah Wharton

Numerade Educator

Problem 4

Let $f:(0, \infty) \rightarrow \mathbb{R}$ satisfy $f(x y)=f(x)+f(y)$ for all $x, y \in(0, \infty) .$ If $f$ is differentiable at 1 , show that $f$ is differentiable at every $c \in(0, \infty)$ and $f^{\prime}(c)=f^{\prime}(1) / c .$ In fact, show that $f$ is infinitely differentiable. If $f^{\prime}(1)=2$, find $f^{(n)}(3) .$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 5

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x+y)=f(x) f(y)$ for all $x, y \in \mathbb{R} .$ If $f$ is differentiable at 0 , then show that $f$ is differentiable at every $c \in \mathbb{R}$ and $f^{\prime}(c)=f^{\prime}(0) f(c) .$ In fact, show that $f$ is infinitely differentiable. If $f^{\prime}(0)=$ 2 , find $f^{(n)}(1)$ for $n \in \mathbb{N}$, in terms of $f(1)$.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 6

Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x+y)=f(x) g(y)+g(x) f(y)$ and $g(x+y)=$
$g(x) g(y)-f(x) f(y)$ for all $x, y \in \mathbb{R} .$ If $f$ and $g$ are differentiable at 0 , then show that $f$ and $g$ are differentiable at every $c \in \mathbb{R}$, and we have $f^{\prime}(c)=g^{\prime}(0) f(c)+f^{\prime}(0) g(c)$ and $g^{\prime}(c)=g^{\prime}(0) g(c)-f^{\prime}(0) f(c) .$ In fact,
show that $f$ and $g$ are infinitely differentiable.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 7

Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x-y)=f(x) g(y)-g(x) f(y)$ and
$g(x-y)=g(x) g(y)+f(x) f(y)$ for all $x, y \in \mathbb{R}$. If $f_{+}^{\prime}(0)$ exists, then show that $f$ and $g$ are differentiable at every $c \in \mathbb{R}$, and $f^{\prime}(c)=f^{\prime}(0) g(c)$ and $g^{\prime}(c)=-f^{\prime}(0) f(c) .$ In fact, show that $f$ and $g$ are infinitely differentiable. If $f_{+}^{\prime}(0)=2$, find $f^{(n)}(1)$ and $g^{(n)}$ (1) in terms of $f(1)$ and $g(1)$. (Hint: Prove that $f$ is an odd function, $g$ is an even function, $f$ and $g$ are differentiable at 0 and $g^{\prime}(0)=0 .$ Use Exercise 6.)

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 8

Find the points on the curve $x^{2}+x y+y^{2}=7$ at which (i) the tangent is parallel to the $x$ -axis, (ii) the tangent is parallel to the $y$ -axis.

KA

Kayla Ashcroft

Numerade Educator

Problem 9

Find the equation of the tangent at $\left(\frac{1}{4}, 4\right)$ to the parametrically defined curve $x(t)=t^{-2}, y(t)=\sqrt{t^{2}+12}$ for $t \in(0.1,1)$.

WM

William Mead

Numerade Educator

Problem 10

Find values of the constants $a, b$, and $c$ for which the graphs of the two functions $f(x)=x^{2}+a x+b$ and $g(x)=x^{3}-c, x \in \mathbb{R}$, intersect at the point $(1,2)$ and have the same tangent there.

Kian Manafi

Numerade Educator

Problem 11

Find the tangents to the implicitly defined curve $x^{2} y+x y^{2}=6$ at points for which $x=1$. Also, compute $\frac{d^{2} u}{d x^{2}}$ at these points.

KA

Kayla Ashcroft

Numerade Educator

Problem 12

Given $n \in \mathbb{N}$, let $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f_{n}(x):=x^{n}$ if $x \geq 0$ and $f_{n}(x):=-x^{n}$ if $x<0 .$ Show that $f_{n}$ is $(n-1)$ -times differentiable on $\mathbb{R}$, $f_{n}^{(n-1)}$ is continuous on $\mathbb{R}$, but $f_{n}^{(n)}(0)$ does not exist.

Nick Johnson

Numerade Educator

Problem 13

Let $D \subseteq \mathbb{R}$ be symmetric about the origin, that is, $-x \in D$ whenever $x \in D .$ If $c \in D$ and $f: D \rightarrow \mathbb{R}$ is either an even or an odd function, then show that the left (hand) derivative $f_{-}^{\prime}(c)$ at $c$ exists if and only if the right (hand) derivative $f_{+}^{\prime}(-c)$ at $-c$ exists. Further, if either (and hence both) of these derivatives exists, then show that $f_{-}^{\prime}(c)=-f_{+}^{\prime}(-c)$ if $f$ is even, and $f_{-}^{\prime}(c)=f_{+}^{\prime}(-c)$ if $f$ is odd. Deduce that if $f$ is differentiable, then $f^{\prime}$ is an odd (resp. even) function according as $f$ is an even (resp. odd) function.

Nick Johnson

Numerade Educator

Problem 14

Let $I$ be an interval, $c \in I$, and $f: I \rightarrow \mathbb{R}$. be any function. Let, as usual, $|f|: I \rightarrow \mathbb{R}$ be the function defined by $|f|(x)=|f(x)|$ for $x \in I$.
(i) Suppose $(c, c+r) \subseteq I$ for some $r>0$ and $f_{+}^{\prime}(c)$ exists. Then show that $|f|_{+}^{\prime}(c)$ exists.
(ii) Suppose If $(c-r, c) \subseteq I$ for some $r>0$ and $f_{-}^{\prime}(c)$ exists. Then show that $|f|_{-}^{\prime}(c)$ exists.
(iii) Suppose $(c-r, c+r) \subseteq I$ for some $r>0$ and $f^{\prime}(c)$ exists. Then show that $|f|^{\prime}(c)$ exists if and only if either there is $\delta>0$ such that $\delta \leq r$ and $f(x)$ has the same sign for all $x \in(c-\delta, c+\delta)$, or $f(c)=f^{\prime}(c)=0$.

Nick Johnson

Numerade Educator

Problem 15

Let $P_{1}=\left(x_{1}, y_{1}\right)$ and $P_{2}=\left(x_{2}, y_{2}\right)$ be two points on the curve $y=$ $a x^{2}+b x+c .$ If $P_{3}=\left(x_{3}, y_{3}\right)$ lies on the arc $P_{1} P_{2}$ and the tangent to the curve at $P_{3}$ is parallel to the chord $P_{1} P_{2}$, show that $x_{3}=\left(x_{1}+x_{2}\right) / 2$.

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 16

Show that the $x$ -axis is a normal to the curve $y^{2}=x$ at $(0,0)$. If three normals can be drawn to this curve from a point $(a, 0)$, show that $a$ must be greater than $\frac{1}{2}$. Find the value of $a$ such that the two normals, other than the $x$ -axis, are perpendicular to each other.

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 17

Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. If $f(a)$ and $f(b)$ are of different signs and $f^{\prime}(x) \neq 0$ for all $x \in(a, b)$, then show that there is a unique $x_{0} \in(a, b)$ such that $f\left(x_{0}\right)=0$.

KA

Kayla Ashcroft

Numerade Educator

Problem 18

Show that the cubic $2 x^{3}+3 x^{2}+6 x+10$ has exactly one real root.

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 19

Let $n \in \mathbb{N}$ and $f:[a, b] \rightarrow \mathbb{R}$ be such that $f^{(n-1)}$ is continuous on $[a, b]$ and $f^{(n)}$ exists in $(a, b) .$ If $f$ vanishes at $n+1$ distinct points in $[a, b]$, then show that $f^{(n)}$ vanishes at least once in $(a, b)$.

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 20

Let $f:\left[-\frac{1}{2}, \frac{1}{2}\right] \rightarrow \mathbb{R}$ be given by
$$
f(x)=\left\{\begin{array}{ll}
\sqrt{2 x-x^{2}} & \text { if } 0 \leq x \leq \frac{1}{2} \\
\sqrt{-2 x-x^{2}} & \text { if }-\frac{1}{2} \leq x \leq 0 .
\end{array}\right.
$$
Show that $f\left(\frac{1}{2}\right)=f\left(-\frac{1}{2}\right)$ but $f^{\prime}(x) \neq 0$ for all $x$ with $0<|x|<\frac{1}{2}$. Does this contradict Rolle's Theorem? Justify your answer.

Vikash Ranjan

Numerade Educator

Problem 21

Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b) .$ If $f(a)<f(b)$, then show that $f^{\prime}(c)>0$ for some $c \in(a, b)$.

KA

Kayla Ashcroft

Numerade Educator

Problem 22

Let $a>0$ and $f:[-a, a] \rightarrow \mathbb{R}$ be continuous. Suppose $f^{\prime}(x)$ exists and $f^{\prime}(x) \leq 1$ for all $x \in(-a, a)$. If $f(a)=a$ and $f(-a)=-a$, then show that $f(x)=x$ for every $x \in(-a, a)$.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 23

In each of the following cases, find a function $f$ that satisfies all the given conditions, or else show that no such function exists.
(i) $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}, f^{\prime}(0)=1, f^{\prime}(1)=1$,
(ii) $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}, f^{\prime}(0)=1, f^{\prime}(1)=2$,
(iii) $f^{\prime \prime}(x) \geq 0$ for all $x \in \mathbb{R}, f^{\prime}(0)=1, f(x) \leq 100$ for all $x>0$,
(iv) $f^{\prime \prime}(x)>0$ for all $x \in \mathbb{R}, f^{\prime}(0)=1, f(x) \leq 1$ for all $x<0$

Nick Johnson

Numerade Educator

Problem 24

Let $f:[a, b] \rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. Suppose $f(a)=a$ and $f(b)=b .$ Show that there is $c \in(a, b)$ such that $f^{\prime}(c)=1 .$ Further, show that there are distinct $c_{1}, c_{2} \in(a, b)$ such that $f^{\prime}\left(c_{1}\right)+f^{\prime}\left(c_{2}\right)=2 .$ More generally, show that for every $n \in \mathbb{N}$, there are $n$ distinct points $c_{1}, \ldots, c_{n} \in(a, b)$ such that $f^{\prime}\left(c_{1}\right)+\cdots+f^{\prime}\left(c_{n}\right)=n$

Hoan Nguyen

Numerade Educator

Problem 25

Let a function $f:[a, b] \rightarrow \mathbb{R}$ be continuous and its second derivative $f^{\prime \prime}$ exist everywhere on the open interval $(a, b) .$ Suppose the line segment joining $(a, f(a))$ and $(b, f(b))$ intersects the graph of $f$ at a third point (c, $f(c))$, where $a<c<b$. Prove that $f^{\prime \prime}(t)=0$ for some $t \in(a, b)$.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 26

Use the MVT to prove that for all $n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ such that $0<a \leq b$, we have $n a^{n-1}(b-a) \leq b^{n}-a^{n} \leq n b^{n-1}(b-a)$.

Michelle Ling

Numerade Educator

Problem 27

Use the MVT to prove that
$$
\frac{1}{3(m+1)^{2 / 3}}<(m+1)^{1 / 3}-m^{1 / 3}<\frac{1}{3 m^{2 / 3}} \quad \text { for all } m \in \mathbb{N}
$$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 28

Use the MVT to prove the following inequalities.
(i) $\frac{27}{16}<\sqrt{3}<\frac{7}{4}$ and $\frac{20}{9}<\sqrt{5}<\frac{9}{4}$.
(ii) $\frac{19}{16}<2^{1 / 3}<\frac{4}{3}, \frac{17}{9}<7^{1 / 3}<\frac{23}{12}$ and $\frac{1298}{625}<9^{1 / 3}<\frac{25}{12}$.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 29

Use the MVT to show that $10.049<\sqrt{101}<10.05$ and $10.24<\sqrt{105}<$ 10.25. Also, find better estimates using Taylor's Theorem with $n=1$, that is, using the Extended MVT.

Nick Johnson

Numerade Educator

Problem 30

Let $f:(a, b) \rightarrow \mathbb{R}$ and $c \in(a, b)$ be such that $f$ is continuous at $c$ and $f^{\prime}(x)$ exist for every $x \in(a, c) \cup(c, b) .$ If $\lim _{x \rightarrow c} f^{\prime}(x)$ exists, then show
that $f^{\prime}(c)$ exists and is equal to this limit.

Nick Johnson

Numerade Educator

Problem 31

(i) Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. If $f(a) \leq g(a)$ and $f^{\prime}(x) \leq g^{\prime}(x)$ for all $x \in(a, b)$, then show that $f(b) \leq g(b)$
(ii) Use (i) to show that $15 x^{2} \leq 8 x^{3}+12 \leq 18 x^{2}$ for all $x \in[1.25,1.5]$. Deduce that the range of the function $h:[1.25,1.5] \rightarrow \mathbb{R}$ given by $h(x)=\left(2 x^{3}+3\right) / 3 x^{2}$ is contained in $[1.25,1.5]$.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 32

Find the $n$ th Taylor polynomial of $f$ around $a$, that is,
$$
P_{n}(x)=f(a)+f^{\prime}(a)(x-a)+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n} \quad \text { for } x \in \mathbb{R}
$$
when $a=0$ and $f(x)$ equals:
(i) $\frac{1}{1-x}$,
(ii) $\frac{1}{1+x}$,
(iii) $\frac{x}{1+x^{2}}$.

Stark Ledbetter

Stark Ledbetter

Numerade Educator

Problem 33

Let $I$ be an interval containing more than one point and $f: I \rightarrow \mathbb{R}$ be any function.
(i) Assume that $f$ is differentiable. If $f^{\prime}$ is nonnegative on $I$ and $f^{\prime}$ vanishes at only a finite number of points on any bounded subinterval of $I$, then show that $f$ is strictly increasing on $I$.
(ii) Assume that $f$ is twice differentiable. If $f^{\prime \prime}$ is nonnegative on $I$ and $f^{\prime \prime}$ vanishes at only a finite number of points on any bounded subinterval of $I$, then show that $f$ is strictly convex on $I$.
(iii) Consider $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=(x-2 n)^{3}+2 n$, where $n \in \mathbb{Z}$ is such that $x \in[2 n-1,2 n+1) .$ Show that $f$ is differentiable on $\mathbb{R}$ and $f^{\prime \prime}$ exists on $(2 n-1,2 n+1)$, but $f_{+}^{\prime \prime}(2 n+1)=6$, whereas $f_{-}^{\prime \prime}(2 n+1)=-6$
for each $n \in \mathbb{N} .$ Also show that $f$ is strictly increasing on $\mathbb{R}$ although $f^{\prime}(2 n)=0$ for each $n \in \mathbb{N}$. (Compare (i) above and Exercise 12 in the list of Revision Exercises at the end of Chapter 7.)
(iv) Consider $g: \mathbb{R} \rightarrow \mathbb{R}$ given by $g(x)=(x-2 n)^{4}+8 n x$, where $n \in \mathbb{Z}$ is such that $x \in[2 n-1,2 n+1) .$ Show that $g$ is twice differentiable on $\mathbb{R}$ and $g^{\prime \prime \prime}$ exists on $(2 n-1,2 n+1)$, but $g_{+}^{\prime \prime \prime}(2 n+1)=24$, whereas
$g_{-}^{\prime \prime \prime}(2 n+1)=-24$ for each $n \in \mathbb{N} .$ Also show that $g$ is strictly convex on $\mathbb{R}$ although $g^{\prime \prime}(2 n)=0$ for each $n \in \mathbb{N}$. (Compare (ii) above and Exercise 13 in the list of Revision Exercises at the end of Chapter $7 .$ )

Aman Gupta

Numerade Educator

Problem 34

Let $I$ be an interval in $\mathbb{R}$ and $c \in I$ be an interior point. If $f: I \rightarrow \mathbb{R}$ is monotonically increasing and if the left and right derivatives of $\bar{f}$ at $c$, namely $f_{-}^{\prime}(c)$ and $f_{+}^{\prime}(c)$, exist, then show that $f_{-}^{\prime}(c) \geq 0$ and $f_{+}^{\prime}(c) \geq 0$. Further, give examples of monotonically increasing functions $f: I \rightarrow \mathbb{R}$ for which $f_{-}^{\prime}(c)<f_{+}^{\prime}(c)$ or for which $f_{-}^{\prime}(c)>f_{+}^{\prime}(c)$.

Nick Johnson

Numerade Educator

Problem 35

Let $f:[a, b] \rightarrow \mathbb{R}$ be such that $f^{\prime}$ is continuous on $[a, b]$ and $f^{\prime \prime}$ exists on $(a, b)$. Show that there is $c \in(a, b)$ such that
$$
f^{\prime \prime}(c)[f(b)-f(a)]=f^{\prime}(c)\left[f^{\prime}(b)-f^{\prime}(a)\right]
$$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 36

Let $f, g, h:[a, b] \rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. Show that there is $c \in(a, b)$ such that the $3 \times 3$ determinant
$$
\left|\begin{array}{lll}
f(a) & f(b) & f^{\prime}(c) \\
g(a) & g(b) & g^{\prime}(c) \\
h(a) & h(b) & h^{\prime}(c)
\end{array}\right|
$$
is zero, that is, $f(a)\left[g(b) h^{\prime}(c)-h(b) g^{\prime}(c)\right]-f(b)\left[g(a) h^{\prime}(c)-h(a) g^{\prime}(c)\right]+$
$f^{\prime}(c)[g(a) h(b)-h(a) g(b)]=0 .$ Deduce that if $h(x)=1$ for all $x \in[a, b]$
we obtain the conclusion of Cauchy's Mean Value Theorem (Proposition $4.36$ ). What does the result say if $g(x)=x$ and $h(x)=1$ for all $x \in[a, b] ?$

Nick Johnson

Numerade Educator

Problem 37

Let $f, g:[a, b] \rightarrow \mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$. If there is $\alpha \in \mathbb{R}$ such that $\left|f^{\prime}(x)\right| \leq \alpha\left|g^{\prime}(x)\right|$ for all $x \in(a, b)$ and if $g^{\prime}(x) \neq 0$ for all $x \in(a, b)$, then show that $|f(b)-f(a)| \leq \alpha|g(b)-g(a)|$. Is the conclusion valid if the condition " $g^{\prime}(x) \neq 0$ for all $x \in(a, b) "$ is omitted?

Nick Johnson

Numerade Educator

Problem 38

Evaluate the following limits:
(i) $\lim _{x \rightarrow 1} \frac{\left(2 x-x^{4}\right)^{1 / 2}-x^{1 / 3}}{1-x^{3 / 4}}$
(ii) $\lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x}{7 x^{2}+1}$,
(iii) $\lim _{x \rightarrow \infty}\left(x-\sqrt{x+x^{2}}\right)$,
(iv) $\lim _{x \rightarrow \infty} \frac{\sqrt{x+2}}{\sqrt{x+1}}$.

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 39

Show that if $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are functions defined by
$$
f(x)=\left\{\begin{array}{ll}
x+2 & \text { if } x \neq 0, \\
0 & \text { if } x=0,
\end{array} \text { and } g(x)=\left\{\begin{array}{ll}
x+1 & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{array}\right.\right.
$$
then
$$
\lim _{x \rightarrow 0} \frac{f^{\prime}(x)}{g^{\prime}(x)}=1 \quad \text { but } \quad \lim _{x \rightarrow 0} \frac{f(x)}{g(x)}=2 .
$$
Does this contradict L'Hôpital's Rule?

Vikash Ranjan

Numerade Educator

Problem 40

Consider the following application of L'Hôpital's Rule:
$$
\lim _{x \rightarrow 1} \frac{3 x^{2}-2 x-1}{x^{2}-x}=\lim _{x \rightarrow 1} \frac{6 x-2}{2 x-1}=\lim _{x \rightarrow 1} \frac{6}{2}=3
$$
Is it correct? Justify.

Vikash Ranjan

Numerade Educator

Problem 41

Consider $f: \mathbb{R} \backslash\{1\} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ defined by
$f(x):=\frac{1}{x-1}$ for $x \neq 1$ and $g(x):=x$ for $x \in \mathbb{R}$.
Show that
$$
\frac{f^{\prime}(x)}{g^{\prime}(x)} \rightarrow-\infty \text { as } x \rightarrow 1^{+}, \quad \text { but } \quad \frac{f(x)}{g(x)} \rightarrow \infty \text { as } x \rightarrow 1^{+} .
$$
Does this contradict L'Hôpital's Rule? Justify.

Vikash Ranjan

Numerade Educator

Problem 42

Let $f:(a, b) \rightarrow \mathbb{R}$ and $c \in(a, b) .$ Show that the following are equivalent:
(i) $f$ is differentiable at $c$.
(ii) There exist $\alpha \in \mathbb{R}, \delta>0$ and a function $\epsilon_{1}:(-\delta, \delta) \rightarrow \mathbb{R}$ such that
$f(c+h)=f(c)+\alpha h+h \epsilon_{1}(h)$ for all $h \in(-\delta, \delta)$ and $\lim _{h \rightarrow 0} \epsilon_{1}(h)=0$
(iii) There exists $\alpha \in \mathbb{R}$ such that
$$
\lim _{h \rightarrow 0} \frac{|f(c+h)-f(c)-\alpha h|}{|h|}=0
$$
If the above conditions hold, then show that $f^{\prime}(c)=\alpha$.

Nick Johnson

Numerade Educator

Problem 43

Let $C$ be an algebraic plane curve, that is, let $C$ be implicitly defined by $F(x, y)=0$, where $F(x, y)$ is a nonzero polynomial in two variables $x$ and $y$ with coefficients in $\mathbb{R} .$ Let the (total) degree of $F(x, y)$ be $n .$ Let $P=\left(x_{0}, y_{0}\right)$ be a point on $C$, so that $F\left(x_{0}, y_{0}\right)=0 .$ (i) If we let $X:=x-c$ and $Y:=y-d$ and define $g(X, Y):=f(x, y)$, then show that $g(X, Y)$ is a polynomial in $X$ and $Y$ with $g(0,0)=0$. Deduce that there is a unique $m \in \mathbb{N}$ such that $m \leq n$ and
$$
g(X, Y)=g_{m}(X, Y)+g_{m+1}(X, Y)+\cdots+g_{n}(X, Y)
$$
where $g_{i}(X, Y)$ is either the zero polynomial or a nonzero homogeneous polynomial of degree $i$, for $m \leq i \leq n$, and $g_{m}(X, Y) \neq 0 .$ We denote the integer $m$ by mult $_{P}(C)$, and call it the multiplicity of $C$ at the point $P$.
(ii) Show that a tangent to the curve $C$ at the point $P$ is defined (as far as calculus is concerned) if and only if mult $_{P}(C)=1$. Moreover, if $\operatorname{mult}_{P}(C)=1$, then there are $\alpha_{1}, \beta_{1} \in \mathbb{R}$ such that $g_{1}(X, Y)=$
$\alpha_{1} X+\beta_{1} Y$, and then the line $\alpha_{1}(x-c)+\beta_{1}(y-d)=0$ is the tangent to $C$ at $P$.
(iii) Show that if $F(x, y)=y-f(x)$ for some polynomial $f(x)$ in one variable $x$, then for the corresponding curve $C$ given by $F(x, y)=0$ we have mult $_{P}(C)=1$ for every $P$ on $C$.
(iv) Determine the integer $m=\operatorname{mult}_{P}(C)$ and a factorization of $g_{m}(X, Y)$ when $P=(0,0)$ and $C$ is the curve implicitly defined by $F(x, y):=$ $y^{2}-x^{2}-x^{3}=0$, or by $F(x, y):=y^{2}-x^{3}=0$
[Note: In view of Exercise 70 of Chapter 1, the initial form $g_{m}(X, Y)$ factors as a product of homogeneous linear polynomials, that is,
$$
g_{m}(X, Y)=\prod_{i=1}^{m}\left(\alpha_{i} X+\beta_{i} Y\right) \text { for some } \alpha_{i}, \beta_{i} \in \mathbb{C}, 1 \leq i \leq m
$$
In the algebraic approach to tangents, the $m$ (complex) lines given by $\alpha_{i}(x-c)+\beta_{i}(y-d)=0$ for $i=1, \ldots, m$, are called the tangent lines to the curve $C$ at the point $P .$ ]

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 44

Let $I$ be an interval and $f: I \rightarrow \mathbb{R}$ be continuous on $I$ and differentiable at every interior point of $I$. If there is a constant $\alpha$ such that $\left|f^{\prime}(x)\right| \leq \alpha$ for all interior points $x$ of $I$, then show that $f$ is uniformly continuous on $I$. Is the converse true? In other words, is it true that if $f: I \rightarrow \mathbb{R}$ is uniformly continuous on $I$ and differentiable at every interior point of $I$, then there is a constant $\alpha$ such that $\left|f^{\prime}(x)\right| \leq \alpha$ for all interior points $x$ of $I ?$

Nick Johnson

Numerade Educator

Problem 45

Let $f:[a, b] \rightarrow \mathbb{R}$ be such that $f^{\prime}$ is continuous on $[a, b]$ and $f^{\prime \prime}$ exists on $(a, b)$. Given any $\xi \in[a, b]$, show that there is $c \in(a, b)$ such that
$$
f(\xi)-f(a)=\frac{f(b)-f(a)}{b-a}(\xi-a)+\frac{f^{\prime \prime}(c)}{2}(\xi-a)(\xi-b)
$$

Nick Johnson

Numerade Educator

Problem 46

Let $f(x)$ be a polynomial. A real number $c$ is called a root of $f(x)$ of multiplicity $m$ if $f(x)=(x-c)^{m} g(x)$ for some polynomial $g(x)$ such that $g(c) \neq 0$.
(i) Let $f(x)$ have $r$ roots (counting multiplicities) in an open interval $(a, b)$. Show that the polynomial $f^{\prime}(x)$ has at least $r-1$ roots in $(a, b)$. Also, give an example where $f^{\prime}(x)$ has more than $r-1$ roots in $(a, b)$. More generally, for $k \in \mathbb{N}$, show that the polynomial $f^{(k)}(x)$ has at least $r-k$ roots in $(a, b)$.
(ii) If $f^{(k)}(x)$ has $s$ roots in $(a, b)$, what can you conclude about the number of roots of $f(x)$ in $(a, b) ?$

Nick Johnson

Numerade Educator

Problem 47

Let $f(x)$ be a polynomial of degree $n$. Given any $a \in \mathbb{R}$, show that
$$
f(x)=f(a)+f^{\prime}(a)(x-a)+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n}, \quad \text { for } x \in \mathbb{R} .
$$
Deduce that $a$ is a root of $f(x)$ of multiplicity $m$ if and only if $f(a)=$ $f^{\prime}(a)=\cdots=f^{(m-1)}(a)=0$ and $f^{(m)}(a) \neq 0 .$ Further, show that if $a$ is a root of $f$ of multiplicity $m$, then
$$
\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h^{m}}=\frac{f^{(m)}(a)}{m !}
$$

Nick Johnson

Numerade Educator

Problem 48

Give an alternative proof of Taylor's Theorem with a single application of Rolle's Theorem by proceeding as follows. Let the notation and hypothesis be as in the statement of Taylor Theorem (Proposition $4.23$ ). Also, as in the proof of Taylor's Theorem, for $x \in[a, b]$, let
$$
P(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n} .
$$
Define $g:[a, b] \rightarrow \mathbb{R}$ by
$$
g(x)=f(x)+f^{\prime}(x)(b-x)+\frac{f^{\prime \prime}(x)}{2 !}(b-x)^{2}+\cdots+\frac{f^{(n)}(x)}{n !}(b-x)^{n}+s(b-x)^{n+1},
$$
where $s=[f(b)-P(b)] /(b-a)^{n+1} .$ Show that $g(a)=g(b)=f(b) .$ Apply Rolle's Theorem to $g$ to deduce Taylor's Theorem.

Nick Johnson

Numerade Educator

Problem 49

Let the notation and hypothesis be as in the statement of Taylor's Theorem (Proposition $4.23$ ). Given any $p \in \mathbb{N}$ with $p \leq n+1$, show that there is $c \in(a, b)$ such that
$$
f(b)=f(a)+f^{\prime}(a)(b-a)+\cdots+\frac{f^{(n)}(a)}{n !}(b-a)^{n}+\frac{f^{(n+1)}(c)}{n ! p}(b-a)^{p}(b-c)^{n-p+1}
$$
[Hint: Proceed as in the previous exercise except to change the $(n+1)$ th power to the $p$ th power in the definitions of $g(x)$ and $s .]$ Show that Taylor's Theorem is a special case of this result with $p=n+1$. Further, show that if $I$ is any interval containing more than one point, $a$ is any point of $I$, and $f: I \rightarrow \mathbb{R}$ is such that $f^{\prime}, f^{\prime \prime}, \ldots, f^{(n)}$ exist on $I$ and $f^{(n+1)}$ exists at every interior point of $I$, then for any $x \in I$, there is $c$ between $a$ and $x$ such that
$f(x)=P_{n}(x)+R_{n, p}(x), \quad$ where $\quad R_{n, p}(x)=\frac{f^{(n+1)}(c)}{n ! p}(x-a)^{p}(x-c)^{n-p+1}$
and where $P_{n}(x)$ is the $n$ th Taylor polynomial of $f$ around $a$. [Note: The remainder term in the above result, namely $R_{n, p}(x)$, is called the Schlömilch form of remainder. It reduces to the Lagrange form of remainder when $p=n+1$, whereas it is called the Cauchy form of remainder when $p=1 .]$

Nick Johnson

Numerade Educator

Problem 50

Let $I$ be an interval containing more than one point and $f: I \rightarrow \mathbb{R}$ be a convex function.
(i) Show that for every interior point $c$ of $I$, both $f_{-}^{\prime}(c)$ and $f_{+}^{\prime}(c)$ exist and $f_{-}^{\prime}(c) \leq f_{+}^{\prime}(c) .$ (Hint: Use Exercise 72 of Chapter $1 .$ )
(ii) Show that for any $x_{1}, x_{2} \in I$ with $x_{1}<x_{2}$, we have $f_{+}^{\prime}\left(x_{1}\right) \leq f_{-}^{\prime}\left(x_{2}\right)$.

Nick Johnson

Numerade Educator

Problem 51

Let $m \in \mathbb{N}$ and $f, g:[a, b] \rightarrow \mathbb{R}$ be such that $f, f^{\prime}, \ldots, f^{(m-1)}$ as well as $g, g^{\prime}, \ldots, g^{(m-1)}$ are continuous on $[a, b]$ and $f^{(m)}, g^{(m)}$ exist on $(a, b)$. Suppose $f^{\prime}(a)=f^{\prime \prime}(a)=\cdots=f^{(m-1)}(a)=0$ and $g^{\prime}(a)=g^{\prime \prime}(a)=\cdots=$
$g^{(m-1)}(a)=0$, but $g^{(m)}(x) \neq 0$ for all $x \in(a, b)$. Prove that there exist $c_{1}, \ldots, c_{m} \in(a, b)$ such that
$$
\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime}\left(c_{1}\right)}{g^{\prime}\left(c_{1}\right)}=\frac{f^{\prime \prime}\left(c_{2}\right)}{g^{\prime \prime}\left(c_{2}\right)}=\cdots=\frac{f^{(m)}\left(c_{m}\right)}{g^{(m)}\left(c_{m}\right)}
$$

Nick Johnson

Numerade Educator

Problem 52

Let $c \in \mathbb{R}, r>0$, and $f:(c-r, c+r) \rightarrow \mathbb{R}$ be such that $f^{\prime \prime}(c)$ exists. Show that
$$
\lim _{h \rightarrow 0^{+}} \frac{f(c+h)+f(c-h)-2 f(c)}{h^{2}}
$$
exists and is equal to $f^{\prime \prime}(c) .$ Give an example of a function that is differentiable on $(c-r, c+r)$, for which this limit exists, but $f^{\prime \prime}(c)$ does not exist.

Nick Johnson

Numerade Educator

Problem 53

Let $c \in \mathbb{R}, r>0, f:(c-r, c+r) \rightarrow \mathbb{R}$, and $n \in \mathbb{N}$ be such that $f^{(n)}(c)$
exists. Show that
$\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)-h f^{\prime}(c)-\cdots-h^{n-1}\left[f^{(n-1)}(c) /(n-1) !\right]}{h^{n}}=\frac{f^{(n)}(c)}{n !}$

Nick Johnson

Numerade Educator