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

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

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

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

Sketch the graph of a differentiable function $y=f(x)$ that hasa. 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 $(3,3)$ .

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$$x=1$$

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The graph of $f^{\prime}$ is given. Determine $x$ -values corresponding to local minima, local maxima, and inflection points for the graph of $f .$

The graph of $f^{\prime}$ is given. Determine $x$-values corresponding to inflection points for the graph of $f .$

The graph of the derivative $f^{\prime}$ of a continuous function $f$ is shown.(a) On what intervals is $f$ increasing or decreasing?(b) At what values of $x$ does $f$ have a local maximum or minimum?(c) On what intervals is $f$ concave upward or downward?(d) State the $x$ -coordinate(s) of the point(s) of inflection.(e) Assuming that $f(0)=0,$ sketch a graph of $f.$

The function $f$ is defined for all $x .$ Use the graph of $f^{\prime}$ to decide: (FIGURE CAN'T COPY)(a) Over what intervals is $f$ increasing? Decreasing?(b) Does $f$ have local maxima or minima? If so, which, and where?

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute maximum at $ 5 $, absolute minimum at $ 2 $, local maximum at $ 3 $, local minima at $ 2 $ and $ 4 $

Graphing $f$ from $f^{\prime}$ Sketch the graph of a continuous function $f$ with $f(0)=-1$ and $f^{\prime}(x)=\left\{\begin{array}{ll}{1,} & {x<-1} \\ {-2,} & {x>-1}\end{array}\right.$

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.$f^{\prime}(x)=0$ at $x=1$ and $3 ; f^{\prime}(2)$ is undefined; $f$ has an absolute maximum at $x=2 ; f$ has neither a local maximum nor a local minimum at $x=1 ;$ and $f$ has an absolute minimum at $x=3.$

Graphing $f$ from $f^{\prime}$ Sketch the graph of a continuous function $f$ with $f(0)=1$ and $f^{\prime}(x)=\left\{\begin{array}{ll}{2,} & {x<2} \\ {-1,} & {x>2}\end{array}\right.$

Sketch a graph of a continuous function $f$ with the following properties:$\cdot f^{\prime}(x)>0$ for all $x$$\cdot f^{\prime \prime}(x)<0$ for $x<2$ and $f^{\prime \prime}(x)>0$ for $x>2$.

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.$f^{\prime}(1)$ and $f^{\prime}(3)$ are undefined; $f^{\prime}(2)=0 ; f$ has a local maximum at $x=1 ; f$ has a local minimum at $x=2 ; f$ has an absolute maximum at $x=3 ;$ and $f$ has an absolute minimum at $x=4.$

Sketch the graph of a continuous function $f$ an [0,4] satisfying the given properties.$f^{\prime}(x)=0$ for $x=1$ and $2 ; f$ has an absolute maximum at $x=4$ f has an absolute minimum at $x=0 ;$ and $f$ has a local minimum at $x=2$

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.$f^{\prime}(x)=0$ for $x=1$ and $2 ; f$ has an absolute maximum at $x=4 ; f$ has an absolute minimum at $x=0 ;$ and $f$ has a local minimum at $x=2.$

Absolute maximum at $ 4 $, absolute minimum at $ 5 $, local maximum at $ 2 $, local minimum at $ 3 $

Absolute minimum at $ 3 $, absolute maximum at $ 4 $, local maximum at $ 2 $

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.$f^{\prime}(x)=0$ for $x=1,2,$ and $3 ; f$ has an absolute minimum at $x=1 ; f$ has no local extremum at $x=2 ;$ and $f$ has an absolute maximum at $x=3.$

Use the graphs of $f^{\prime}$ and $f^{\prime \prime}$ to find the critical points and inflection points of $f$, the intervals on which $f$ is increasing or decreasing, and the intervals of concavity. Then graph $f$ assuming $f(0)=0$.(GRAPH CAN'T COPY)

Graphing $f^{\prime}$ from $f$ Given the graph of the function $f$ below,sketch a graph of the derivative of $f .$

The graph of $f^{\prime}$ is given. Assume that $f(0)=1$ and sketch a possible continuous graph of $f$.Graph cannot copy

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = | x | $

Use the graphs shown here, which include all extrema.(GRAPH CANT COPY)Give the local and absolute extreme values of $f .$

$ f(x) = e^x $

$9-12$ Sketch the graph of a function $f$ that is continuous on $[1,5]$and has the given properties.

Absolute maximum at 5, absolute minimum at 2,local maximum at 3, local minima at 2 and 4

The graph of the derivative $ f' $ of a continuous function $ f $ is shown.(a) On what intervals is $ f $ increasing? Decreasing?(b) At what values of $ x $ does $ f $ have a local maximum? Local minimum?(c) On what intervals is $ f $ concave upward? Concave downward?(d) State the $ x $-coordinate(s) of the point(s) of inflection.(e) Assuming that $ f(0) = 0 $, sketch a graph of $ f $.

Find the intervals on which $f$ is increasing and decreasing. Superimpose the graphs of $f$ and $f^{\prime}$ to verify your work.$$f(x)=x^{3}+4 x$$

Sketch a continuous function $f$ on some interval that has the properties described. Answers will vary.The function $f$ has one inflection point but no local extrema.

Sketch a graph of a function $f$ that is continuous on $(-\infty, \infty)$ and has the following properties.$$f^{\prime}(x) > 0, f^{\prime \prime}(x) > 0$$

Use the derivative $f^{\prime}$ to determine the $x$ -coordinates of the local maxima and minima of $f$, and the intervals on which $f$ is increasing or decreasing. Sketch a possible graph of$f(f$ is not unique).$$f^{\prime}(x)=(x-1)(x+2)(x+4)$$

$7-10=$ Sketch the graph of a function $f$ that is continuouson $[1,5]$ and has the given properties.Absolute maximum at $5,$ absolute minimum at $2,$local maximum at $3,$ local minima at 2 and 4

$7-10=$ Sketch the graph of a function $f$ that is continuouson $[1,5]$ and has the given properties.$f$ has no local maximum or minimum, but 2 and 4 arecritical numbers

Show the derivative $f^{\prime}$ of $f$.(a) Where is $f$ increasing and where is $f$ decreasing? What are the $x$ -coordinates of the local maxima and minima of $f ?$.(b) Sketch a possible graph for $f$. (You don't need a scale on the vertical axis.)(FIGURE CAN'T COPY)

$9-12$ Sketch the graph of a function $f$ that is continuous on $[1,5]$and has the given properties.$f$ has no local maximum or minimum, but 2 and 4 are critical numbers

$ f(x) = 1/x $, $ 1 < x < 3 $

Absolute minimum at 1, absolute maximum at 5,local maximum at 2, local minimum at 4

$ f(x) = \ln x $, $ 0 < x \leqslant 2 $

The graph of a function $f$ is given. Use the graph to find:(a) The numbers, if any, at which $f$ has a local maximum. What are the local maximum values?(b) The numbers, if any, at which $f$ has a local minimum. What are the local minimum values?

(Check your book for graph)

$ f(x) = 1/x $, $ x \geqslant 1 $

Sketch the graph of $f$ by hand and use your sketch to find the absolute and local maximum and minimum values of $f .$ (Use the graphs and transformations of Sections 1.2 and $1.3 .$ )$f(x)=8-3 x, \quad x \geqslant 1$

$ f(x) = \frac{1}{2}(3x-1) $, $ x \leqslant 3 $

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## Recommended Questions

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

The graph of $f^{\prime}$ is given. Determine $x$ -values corresponding to local minima, local maxima, and inflection points for the graph of $f .$

The graph of $f^{\prime}$ is given. Determine $x$-values corresponding to inflection points for the graph of $f .$

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

(a) On what intervals is $f$ increasing or decreasing?

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

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

(d) State the $x$ -coordinate(s) of the point(s) of inflection.

(e) Assuming that $f(0)=0,$ sketch a graph of $f.$

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

(a) On what intervals is $f$ increasing or decreasing?

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

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

(d) State the $x$ -coordinate(s) of the point(s) of inflection.

(e) Assuming that $f(0)=0,$ sketch a graph of $f.$

The function $f$ is defined for all $x .$ Use the graph of $f^{\prime}$ to decide: (FIGURE CAN'T COPY)

(a) Over what intervals is $f$ increasing? Decreasing?

(b) Does $f$ have local maxima or minima? If so, which, and where?

The function $f$ is defined for all $x .$ Use the graph of $f^{\prime}$ to decide: (FIGURE CAN'T COPY)

(a) Over what intervals is $f$ increasing? Decreasing?

(b) Does $f$ have local maxima or minima? If so, which, and where?

The function $f$ is defined for all $x .$ Use the graph of $f^{\prime}$ to decide: (FIGURE CAN'T COPY)

(a) Over what intervals is $f$ increasing? Decreasing?

(b) Does $f$ have local maxima or minima? If so, which, and where?

(a) Over what intervals is $f$ increasing? Decreasing?

(b) Does $f$ have local maxima or minima? If so, which, and where?

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute maximum at $ 5 $, absolute minimum at $ 2 $, local maximum at $ 3 $, local minima at $ 2 $ and $ 4 $

Graphing $f$ from $f^{\prime}$ Sketch the graph of a continuous function $f$ with $f(0)=-1$ and $f^{\prime}(x)=\left\{\begin{array}{ll}{1,} & {x<-1} \\ {-2,} & {x>-1}\end{array}\right.$

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.

$f^{\prime}(x)=0$ at $x=1$ and $3 ; f^{\prime}(2)$ is undefined; $f$ has an absolute maximum at $x=2 ; f$ has neither a local maximum nor a local minimum at $x=1 ;$ and $f$ has an absolute minimum at $x=3.$

Graphing $f$ from $f^{\prime}$ Sketch the graph of a continuous function $f$ with $f(0)=1$ and $f^{\prime}(x)=\left\{\begin{array}{ll}{2,} & {x<2} \\ {-1,} & {x>2}\end{array}\right.$

Sketch a graph of a continuous function $f$ with the following properties:

$\cdot f^{\prime}(x)>0$ for all $x$

$\cdot f^{\prime \prime}(x)<0$ for $x<2$ and $f^{\prime \prime}(x)>0$ for $x>2$.

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.

$f^{\prime}(1)$ and $f^{\prime}(3)$ are undefined; $f^{\prime}(2)=0 ; f$ has a local maximum at $x=1 ; f$ has a local minimum at $x=2 ; f$ has an absolute maximum at $x=3 ;$ and $f$ has an absolute minimum at $x=4.$

Sketch the graph of a continuous function $f$ an [0,4] satisfying the given properties.

$f^{\prime}(x)=0$ for $x=1$ and $2 ; f$ has an absolute maximum at $x=4$ f has an absolute minimum at $x=0 ;$ and $f$ has a local minimum at $x=2$

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.

$f^{\prime}(x)=0$ for $x=1$ and $2 ; f$ has an absolute maximum at $x=4 ; f$ has an absolute minimum at $x=0 ;$ and $f$ has a local minimum at $x=2.$

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute maximum at $ 4 $, absolute minimum at $ 5 $, local maximum at $ 2 $, local minimum at $ 3 $

Sketch the graph of a function $ f $ that is continuous on $ [1, 5] $ and has the given properties.

Absolute minimum at $ 3 $, absolute maximum at $ 4 $, local maximum at $ 2 $

Sketch the graph of a continuous function $f$ on $[0,4]$ satisfying the given properties.

$f^{\prime}(x)=0$ for $x=1,2,$ and $3 ; f$ has an absolute minimum at $x=1 ; f$ has no local extremum at $x=2 ;$ and $f$ has an absolute maximum at $x=3.$

Use the graphs of $f^{\prime}$ and $f^{\prime \prime}$ to find the critical points and inflection points of $f$, the intervals on which $f$ is increasing or decreasing, and the intervals of concavity. Then graph $f$ assuming $f(0)=0$.

(GRAPH CAN'T COPY)

Graphing $f^{\prime}$ from $f$ Given the graph of the function $f$ below,

sketch a graph of the derivative of $f .$

The graph of $f^{\prime}$ is given. Assume that $f(0)=1$ and sketch a possible continuous graph of $f$.

Graph cannot copy

The graph of $f^{\prime}$ is given. Assume that $f(0)=1$ and sketch a possible continuous graph of $f$.

Graph cannot copy

The graph of $f^{\prime}$ is given. Assume that $f(0)=1$ and sketch a possible continuous graph of $f$.

Graph cannot copy

Graph cannot copy

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = | x | $

Use the graphs shown here, which include all extrema.

(GRAPH CANT COPY)

Give the local and absolute extreme values of $f .$

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = e^x $

$9-12$ Sketch the graph of a function $f$ that is continuous on $[1,5]$

and has the given properties.

Absolute maximum at 5, absolute minimum at 2,

local maximum at 3, local minima at 2 and 4

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

(a) On what intervals is $ f $ increasing? Decreasing?

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

(c) On what intervals is $ f $ concave upward? Concave downward?

(d) State the $ x $-coordinate(s) of the point(s) of inflection.

(e) Assuming that $ f(0) = 0 $, sketch a graph of $ f $.

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

(a) On what intervals is $ f $ increasing? Decreasing?

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

(c) On what intervals is $ f $ concave upward? Concave downward?

(d) State the $ x $-coordinate(s) of the point(s) of inflection.

(e) Assuming that $ f(0) = 0 $, sketch a graph of $ f $.

Find the intervals on which $f$ is increasing and decreasing. Superimpose the graphs of $f$ and $f^{\prime}$ to verify your work.

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

Sketch a continuous function $f$ on some interval that has the properties described. Answers will vary.

The function $f$ has one inflection point but no local extrema.

Sketch a graph of a function $f$ that is continuous on $(-\infty, \infty)$ and has the following properties.

$$f^{\prime}(x) > 0, f^{\prime \prime}(x) > 0$$

Use the derivative $f^{\prime}$ to determine the $x$ -coordinates of the local maxima and minima of $f$, and the intervals on which $f$ is increasing or decreasing. Sketch a possible graph of

$f(f$ is not unique).

$$f^{\prime}(x)=(x-1)(x+2)(x+4)$$

$7-10=$ Sketch the graph of a function $f$ that is continuous

on $[1,5]$ and has the given properties.

Absolute maximum at $5,$ absolute minimum at $2,$

local maximum at $3,$ local minima at 2 and 4

$7-10=$ Sketch the graph of a function $f$ that is continuous

on $[1,5]$ and has the given properties.

$f$ has no local maximum or minimum, but 2 and 4 are

critical numbers

Show the derivative $f^{\prime}$ of $f$.

(a) Where is $f$ increasing and where is $f$ decreasing? What are the $x$ -coordinates of the local maxima and minima of $f ?$.

(b) Sketch a possible graph for $f$. (You don't need a scale on the vertical axis.)

(FIGURE CAN'T COPY)

Show the derivative $f^{\prime}$ of $f$.

(a) Where is $f$ increasing and where is $f$ decreasing? What are the $x$ -coordinates of the local maxima and minima of $f ?$.

(b) Sketch a possible graph for $f$. (You don't need a scale on the vertical axis.)

(FIGURE CAN'T COPY)

$9-12$ Sketch the graph of a function $f$ that is continuous on $[1,5]$

and has the given properties.

$f$ has no local maximum or minimum, but 2 and 4 are critical numbers

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1/x $, $ 1 < x < 3 $

$9-12$ Sketch the graph of a function $f$ that is continuous on $[1,5]$

and has the given properties.

Absolute minimum at 1, absolute maximum at 5,

local maximum at 2, local minimum at 4

$ f(x) = \ln x $, $ 0 < x \leqslant 2 $

The graph of a function $f$ is given. Use the graph to find:

(a) The numbers, if any, at which $f$ has a local maximum. What are the local maximum values?

(b) The numbers, if any, at which $f$ has a local minimum. What are the local minimum values?

(Check your book for graph)

The graph of a function $f$ is given. Use the graph to find:

(a) The numbers, if any, at which $f$ has a local maximum. What are the local maximum values?

(b) The numbers, if any, at which $f$ has a local minimum. What are the local minimum values?

(Check your book for graph)

The graph of a function $f$ is given. Use the graph to find:

(a) The numbers, if any, at which $f$ has a local maximum. What are the local maximum values?

(b) The numbers, if any, at which $f$ has a local minimum. What are the local minimum values?

(Check your book for graph)

(a) The numbers, if any, at which $f$ has a local maximum. What are the local maximum values?

(b) The numbers, if any, at which $f$ has a local minimum. What are the local minimum values?

(Check your book for graph)

$ f(x) = 1/x $, $ x \geqslant 1 $

Sketch the graph of $f$ by hand and use your sketch to find the absolute and local maximum and minimum values of $f .$ (Use the graphs and transformations of Sections 1.2 and $1.3 .$ )

$f(x)=8-3 x, \quad x \geqslant 1$

$ f(x) = \frac{1}{2}(3x-1) $, $ x \leqslant 3 $