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

$$

## Discussion

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## Recommended Questions

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 or minimum values?

$$f^{\prime}(x)=(\sin x-1)(2 \cos x+1), 0 \leq x \leq 2 \pi$$

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

$$

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 or minimum values?

$$f^{\prime}(x)=(x-1) e^{-x}$$

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 or minimum values?

$$f^{\prime}(x)=(x-1)^{2}(x+2)$$

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 or minimum values?

$$f^{\prime}(x)=(x-7)(x+1)(x+5)$$

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 or minimum values?

$$f^{\prime}(x)=(x-1)^{2}(x+2)^{2}$$

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)

$$

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 or minimum values?

$$f^{\prime}(x)=x(x-1)$$

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 or minimum values?

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

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 or minimum values?

$$f^{\prime}(x)=(x-1)(x+2)$$

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

$$

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)

$$

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 or minimum values?

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

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 or minimum values?

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

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}

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}

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}

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}

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}

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}

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}

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}

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}

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

$$

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=\cos ^{2} x-2 \sin x, \quad 0 \leqslant x \leqslant 2 \pi$

(a) Find the intervals on which $ f $ is increasing or decreasing.

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = \sin x + \cos x $, $ 0 \leqslant x \leqslant 2\pi $

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=\sin x+\cos x, \quad 0 \leqslant x \leqslant 2 \pi$

(a) Find the intervals on which $ f $ is increasing or decreasing.

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = \cos^2 x - 2\sin x $, $ 0 \leqslant x \leqslant 2\pi $

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

$$

(a) Find the intervals on which $ f $ is increasing or decreasing.

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = x^2 \ln x $

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=x^{4}-2 x^{2}+3$

a. Locate the critical points of $f$

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

$$f(x)=\sqrt{x} \ln x ;(0, \infty)$$

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = x^2 - x - \ln x $

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}

$$

a. Find the critical points of the following functions on the given interval.

b. Use a graphing device to determine whether the critical points correspond to local maxima, local minima, or neither.

c. Find the absolute maximum and minimum values on the given interval when they exist.

$$f(\theta)=2 \sin \theta+\cos \theta ;[-2 \pi, 2 \pi]$$

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=x^{2} \ln x$

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

$$

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}

$$

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=x^{2}-x-\ln x$

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = e^{2x} + e^{-x} $

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = x^4e^{-x} x $

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = \frac{x}{x^2 + 1} $

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=e^{2 x}+e^{-x}$

a. Locate the critical points of $f$

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

$$f(x)=2 x^{3}+3 x^{2}-12 x+1 ;[-2,4]$$

a. Locate the critical points of $f$

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

$$f(x)=x \sqrt{4-x^{2}} \text { on }[-2,2]$$

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = x^4 - 2x^2 + 3 $

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=x^{4} e^{-x}$

(b) Find the local maximum and minimum values of $ f $.

(c) Find the intervals of concavity and the inflection points.

$ f(x) = 2x^3 - 9x^2 + 12x - 3 $

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=4 x^{3}+3 x^{2}-6 x+1$

Critical points and extreme values

a. Find the critical points of the following functions on the given interval. Use a root finder, if necessary.

b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.

E. Find the absolute maximum and minimum values on the given inter. wal, if they exist.

$$f(\theta)=2 \sin \theta+\cos \theta \text { on }[-2 \pi, 2 \pi]$$

The graph of the first derivative $ f' $ of a function $ f $ is shown.

(a) On what intervals is $ f $ increasing? Explain.

(b) At what values of $ x $ does $ f $ have a local maximum or minimum? Explain.

(c) On what intervals is $ f $ concave upward or concave downward? Explain.

(d) What are the $ x $-coordinates of the inflection points of $ f $? Why?

a. Locate the critical points of $f$c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

b. Use the First Derivative Test to locate the local maximum and minimum values.

$$f(x)=-x^{2}-x+2 ;[-4,4]$$

Do the following:

(a) Find $f^{\prime}$ and $f^{\prime \prime}$.

(b) Find the critical points of $f$.

(c) Find any inflection points of $f$.

(d) Evaluate $f$ at its critical points and at the endpoints of the given interval. Identify local and global maxima and minima of $f$ in the interval.

(e) Graph $f$.

$$f(x)=x+\sin x \quad(0 \leq x \leq 2 \pi)$$

$11-20$

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f$ .

(c) Find the intervals of concavity and the inflection points.

$$f(x)=\cos ^{2} x-2 \sin x, \quad 0 \leqq x \leqslant 2 \pi$$

a. Find the critical points of the following functions on the domain or on the given interval.

b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither.

$$f(x)=x-\tan ^{-1} x$$

The graph of the first derivative $f^{\prime}$ of a function $f$ is shown.

(a) On what intervals is $f$ increasing? Explain.

(b) At what values of $x$ does $f$ have a local maximum or minimum? Explain.

(c) On what intervals is $f$ concave upward or concave downward? Explain.

(d) What are the $x$ -coordinates of the inflection points of $f ?$ Why?

a. Find the critical points of the following functions on the domain or on the given interval.

b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither.

$$f(x)=\sin x \cos x \text { on } [0,2 \pi]$$

a. Find the critical points of $f$ on the given interval.

b. Determine the absolute extreme values of $f$ on the given interval when they exist.

c. Use a graphing utility to confirm your conclusions.

$$f(x)=\cos ^{2} x \text { on } [0, \pi]$$

Suppose the derivative of $f$ is $f^{\prime}(x)=(x-1)(x-2)$

a. Find the critical points of $f$

b. On what intervals is $f$ increasing and on what intervals is $f$ decreasing?

(a) Find the intervals on which $f$ is increasing or decreasing.

(b) Find the local maximum and minimum values of $f .$

(c) Find the intervals of concavity and the inflection points.

$f(x)=2 x^{3}+3 x^{2}-36 x$

a. Find the critical points of the following functions on the given interval.

b. Use a graphing device to determine whether the critical points correspond to local maxima, local minima, or neither.

c. Find the absolute maximum and minimum values on the given interval when they exist.

$$f(t)=3 t /\left(t^{2}+1\right) ;[-2,2]$$

Suppose the derivative of $f$ is $f^{\prime}(x)=x-3$

a. Find the critical points of $f$

b. On what intervals is $f$ increasing and on what intervals is $f$ decreasing?