Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=x(x-1)\end{equation}

Runpeng L.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=(x-1)(x+2)\end{equation}

Matt J.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=(x-1)^{2}(x+2)\end{equation}

Runpeng L.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=(x-1)^{2}(x+2)^{2}\end{equation}

Matt J.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=(x-1)(x+2)(x-3)\end{equation}

Runpeng L.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=(x-7)(x+1)(x+5)\end{equation}

Matt J.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=\frac{x^{2}(x-1)}{x+2}, \quad x \neq-2\end{equation}

Runpeng L.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=\frac{(x-2)(x+4)}{(x+1)(x-3)}, \quad x \neq-1,3\end{equation}

Matt J.

Numerade Educator

Answer the following questions about the functions whose derivatives are given:

\begin{equation}\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}\end{equation}

\begin{equation}f^{\prime}(x)=1-\frac{4}{x^{2}}, \quad x \neq 0\end{equation}

Runpeng L.

Numerade Educator

Answer the following questions about the functions whose derivatives

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.

Numerade Educator

Answer the following questions about the functions whose derivatives

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.

Numerade Educator

Answer the following questions about the functions whose derivatives

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.

Numerade Educator

Answer the following questions about the functions whose derivatives

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

$$

Check back soon!

Answer the following questions about the functions whose derivatives

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.

Numerade Educator

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.

Numerade Educator

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Matt J.

Numerade Educator

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Runpeng L.

Numerade Educator

b. Identify the function's local and absolute extreme values, if any, saying where they occur.

Matt J.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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)

$$

Alvar G.

Numerade Educator

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.

Numerade Educator

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)

$$

Alvar G.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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.

Numerade Educator

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

$$

Willis J.

Numerade Educator

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.

Numerade Educator

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

$$

Alvar G.

Numerade Educator

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.

Numerade Educator

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

$$

Romaila S.

Numerade Educator

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.

Numerade Educator

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

$$

Alvar G.

Numerade Educator

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.

Numerade Educator

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

$$

Alvar G.

Numerade Educator

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.

Numerade Educator

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

$$

Check back soon!

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.

Numerade Educator

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

Check back soon!

Matt J.

Numerade Educator

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

$$

Vanessa R.

Numerade Educator

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.

Numerade Educator

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}

$$

Alvar G.

Numerade Educator

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.

Numerade Educator

Sketch the graph of a continuous function $y=g(x)$ such that

$$\begin{array}{c}{\text { a. } g(2)=2,0 < g^{\prime}<1 \text { for } x < 2, g^{\prime}(x) \rightarrow 1^{-} \text { as } x \rightarrow 2^{-}} \\ {-1< g^{\prime} < 0 \text { for } x > 2, \text { and } g^{\prime}(x) \rightarrow-1^{+} \text { as } x \rightarrow 2^{+}}\end{array}$$

$$\begin{array}{l}{\text { b. } g(2)=2, g^{\prime}<0 \text { for } x<2, g^{\prime}(x) \rightarrow-\infty \text { as } x \rightarrow 2^{-}} \\ {g^{\prime}>0 \text { for } x>2, \text { and } g^{\prime}(x) \rightarrow \infty \text { as } x \rightarrow 2^{+} \text { . }}\end{array}$$

Check back soon!

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.

Numerade Educator

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 $2.3 .$ .

Check back soon!

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.

Numerade Educator

Determine the values of constants $a$ and $b$ so that $f(x)=a x^{2}+b x$

has an absolute maximum at the point $(1,2) .$

Raia O.

Numerade Educator

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.

Numerade Educator