# Thomas Calculus

## Educators

Problem 1

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=x(x-1)$$

Runpeng L.

Problem 2

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=(x-1)(x+2)$$

Matt J.

Problem 3

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=(x-1)^{2}(x+2)$$

Runpeng L.

Problem 4

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=(x-1)^{2}(x+2)^{2}$$

Matt J.

Problem 5

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=(x-1)(x+2)(x-3)$$

Runpeng L.

Problem 6

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=(x-7)(x+1)(x+5)$$

Matt J.

Problem 7

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=\frac{x^{2}(x-1)}{x+2}, \quad x \neq-2$$

Runpeng L.

Problem 8

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=\frac{(x-2)(x+4)}{(x+1)(x-3)}, \quad x \neq-1,3$$

Matt J.

Problem 9

$$\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}$$
$$f^{\prime}(x)=1-\frac{4}{x^{2}}, \quad x \neq 0$$

Runpeng L.

Problem 10

are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$f^{\prime}(x)=3-\frac{6}{\sqrt{x}}, \quad x \neq 0$$

Matt J.

Problem 11

are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$f^{\prime}(x)=x^{-1 / 3}(x+2)$$

Runpeng L.

Problem 12

are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$f^{\prime}(x)=x^{-1 / 2}(x-3)$$

Matt J.

Problem 13

are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$f^{\prime}(x)=(\sin x-1)(2 \cos x+1), 0 \leq x \leq 2 \pi$$

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Problem 14

are given.
a. What are the critical points of $f ?$
b. On what open intervals is $f$ increasing or decreasing?
c. At what points, if any, does $f$ assume local maximum and minimum values?
$$f^{\prime}(x)=(\sin x+\cos x)(\sin x-\cos x), 0 \leq x \leq 2 \pi$$

Matt J.

Problem 15

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Runpeng L.

Problem 16

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Matt J.

Problem 17

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Runpeng L.

Problem 18

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Matt J.

Problem 19

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$g(t)=-t^{2}-3 t+3$$

Runpeng L.

Problem 20

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$g(t)=-3 t^{2}+9 t+5$$

Matt J.

Problem 21

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$h(x)=-x^{3}+2 x^{2}$$

Runpeng L.

Problem 22

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$h(x)=2 x^{3}-18 x$$

Matt J.

Problem 23

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(\theta)=3 \theta^{2}-4 \theta^{3}$$

Runpeng L.

Problem 24

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(\theta)=6 \theta-\theta^{3}$$

Matt J.

Problem 25

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(r)=3 r^{3}+16 r$$

Runpeng L.

Problem 26

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$h(r)=(r+7)^{3}$$

Matt J.

Problem 27

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(x)=x^{4}-8 x^{2}+16$$

Runpeng L.

Problem 28

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$g(x)=x^{4}-4 x^{3}+4 x^{2}$$

Matt J.

Problem 29

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$H(t)=\frac{3}{2} t^{4}-t^{6}$$

Runpeng L.

Problem 30

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$K(t)=15 t^{3}-t^{5}$$

Matt J.

Problem 31

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(x)=x-6 \sqrt{x-1}$$

Runpeng L.

Problem 32

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$g(x)=4 \sqrt{x}-x^{2}+3$$

Matt J.

Problem 33

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$g(x)=x \sqrt{8-x^{2}}$$

Runpeng L.

Problem 34

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$g(x)=x^{2} \sqrt{5-x}$$

Matt J.

Problem 35

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(x)=\frac{x^{2}-3}{x-2}, \quad x \neq 2$$

Runpeng L.

Problem 36

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(x)=\frac{x^{3}}{3 x^{2}+1}$$

Matt J.

Problem 37

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$f(x)=x^{1 / 3}(x+8)$$

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Problem 38

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$g(x)=x^{2 / 3}(x+5)$$

Matt J.

Problem 39

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.
$$h(x)=x^{1 / 3}\left(x^{2}-4\right)$$

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Problem 40

a. Find the open intervals on which the function is increasing and decreasing.
b. Identify the function's local and absolute extreme values, if any, saying where they occur.$$k(x)=x^{2 / 3}\left(x^{2}-4\right)$$

Matt J.

Problem 41

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$f(x)=2 x-x^{2}, \quad-\infty<x \leq 2$$

Runpeng L.

Problem 42

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$f(x)=(x+1)^{2}, \quad-\infty<x \leq 0$$

Matt J.

Problem 43

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$g(x)=x^{2}-4 x+4, \quad 1 \leq x<\infty$$

Runpeng L.

Problem 44

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$g(x)=-x^{2}-6 x-9, \quad-4 \leq x<\infty$$

Matt J.

Problem 45

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$f(t)=12 t-t^{3}, \quad-3 \leq t<\infty$$

Runpeng L.

Problem 46

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$f(t)=t^{3}-3 t^{2}, \quad-\infty< t \leq 3$$

Matt J.

Problem 47

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$h(x)=\frac{x^{3}}{3}-2 x^{2}+4 x, \quad 0 \leq x<\infty$$

Runpeng L.

Problem 48

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$k(x)=x^{3}+3 x^{2}+3 x+1, \quad-\infty<x \leq 0$$

Matt J.

Problem 49

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$f(x)=\sqrt{25-x^{2}}, \quad-5 \leq x \leq 5$$

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Problem 50

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$f(x)=\sqrt{x^{2}-2 x-3}, \quad 3 \leq x<\infty$$

Matt J.

Problem 51

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$g(x)=\frac{x-2}{x^{2}-1}, \quad 0 \leq x<1$$

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Problem 52

a. Identify the function's local extreme values in the given domain, and say where they occur.
b. Which of the extreme values, if any, are absolute?
c. Support your findings with a graphing calculator or computer grapher.
$$g(x)=\frac{x^{2}}{4-x^{2}}, \quad-2< x \leq 1$$

Matt J.

Problem 53

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=\sin 2 x, \quad 0 \leq x \leq \pi$$

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Problem 54

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=\sin x-\cos x, \quad 0 \leq x \leq 2 \pi$$

Matt J.

Problem 55

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=\sqrt{3} \cos x+\sin x, \quad 0 \leq x \leq 2 \pi$$

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Problem 56

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=-2 x+\tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}$$

Matt J.

Problem 57

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=\frac{x}{2}-2 \sin \frac{x}{2}, \quad 0 \leq x \leq 2 \pi$$

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Problem 58

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=-2 \cos x-\cos ^{2} x, \quad-\pi \leq x \leq \pi$$

Matt J.

Problem 59

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=\csc ^{2} x-2 \cot x, \quad 0<x<\pi$$

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Problem 60

a. Find the local extrema of each function on the given interval,
and say where they occur.
b. Graph the function and its derivative together. Comment on
the behavior of $f$ in relation to the signs and values of $f^{\prime}.$
$$f(x)=\sec ^{2} x-2 \tan x, \quad \frac{-\pi}{2}< x <\frac{\pi}{2}$$

Matt J.

Problem 61

The graph of $f^{\prime}$ is given. Assume that $f$ is continuous and determine the $x$ -values corresponding to local minima and local maxima.

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Problem 62

The graph of $f^{\prime}$ is given. Assume that $f$ is continuous and determine the $x$ -values corresponding to local minima and local maxima.

Matt J.

Problem 63

Show that the functions have local extreme values at the given values of $\theta,$ and say which kind of local extreme the function has.
$$h(\theta)=3 \cos \frac{\theta}{2}, \quad 0 \leq \theta \leq 2 \pi, \quad \text { at } \theta=0 \text { and } \theta=2 \pi$$

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Problem 64

Show that the functions have local extreme values at the given values of $\theta,$ and say which kind of local extreme the function has.
$$h(\theta)=5 \sin \frac{\theta}{2}, \quad 0 \leq \theta \leq \pi, \quad \text { at } \theta=0 \text { and } \theta=\pi$$

Matt J.

Problem 65

Sketch the graph of a differentiable function $y=f(x)$ through the
point $(1,1)$ if $f^{\prime}(1)=0$ and
$$\begin{array}{l}{\text { a. } f^{\prime}(x)>0 \text { for } x<1 \text { and } f^{\prime}(x)<0 \text { for } x>1} \\ {\text { b. } f^{\prime}(x)<0 \text { for } x<1 \text { and } f^{\prime}(x)>0 \text { for } x>1} \\ {\text { c. } f^{\prime}(x)>0 \text { for } x \neq 1} \\ {\text { d. } f^{\prime}(x)<0 \text { for } x \neq 1}\end{array}$$

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Problem 66

Sketch the graph of a differentiable function $y=f(x)$ that has
a. a local minimum at $(1,1)$ and a local maximum at $(3,3)$
b. a local maximum at $(1,1)$ and a local minimum at $(3,3)$
c. 1 ocal maxima at $(1,1)$ and $(3,3)$
d. local minima at $(1,1)$ and $(3,3)$ .

Matt J.

Problem 67

Sketch the graph of a continuous function $y=g(x)$ such that
$$\begin{array}{c}{\text { a. } g(2)=2,0 < g^{\prime}<1 \text { for } x < 2, g^{\prime}(x) \rightarrow 1^{-} \text { as } x \rightarrow 2^{-}} \\ {-1< g^{\prime} < 0 \text { for } x > 2, \text { and } g^{\prime}(x) \rightarrow-1^{+} \text { as } x \rightarrow 2^{+}}\end{array}$$
$$\begin{array}{l}{\text { b. } g(2)=2, g^{\prime}<0 \text { for } x<2, g^{\prime}(x) \rightarrow-\infty \text { as } x \rightarrow 2^{-}} \\ {g^{\prime}>0 \text { for } x>2, \text { and } g^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 2^{+} \text { . }}\end{array}$$

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Problem 68

Sketch the graph of a continuous function $y=h(x)$ such that
$$\begin{array}{l}{\text { a. } h(0)=0,-2 \leq h(x) \leq 2 \text { for all } x, h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{-}} \\ {\text { and } h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{+}} \\ {\text { b. } h(0)=0,-2 \leq h(x) \leq 0 \text { for all } x, h^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 0^{-}} \\ {\quad \text { and } h^{\prime}(x) \rightarrow-\infty \text { as } x \rightarrow 0^{+} \text { . }}\end{array}$$

Matt J.

Problem 69

Discuss the extreme-value behavior of the function $f(x)=$ $x \sin (1 / x), x \neq 0 .$ How many critical points does this function have? Where are they located on the $x$ -axis? Does $f$ have an absolute minimum? An absolute maximum? (See Exercise 49 in Section $2.3 .$ .

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Problem 70

Find the open intervals on which the function $f(x)=a x^{2}+$ $b x+c, a \neq 0,$ is increasing and decreasing. Describe the reasoning behind your answer.

Matt J.

Problem 71

Determine the values of constants $a$ and $b$ so that $f(x)=a x^{2}+b x$
has an absolute maximum at the point $(1,2) .$

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Problem 72

Determine the values of constants $a, b, c,$ and $d$ so that
$f(x)=a x^{3}+b x^{2}+c x+d$ has a local maximum at the point
$(0,0)$ and a local minimum at the point $(1,-1)$ .

Matt J.