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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## Recommended Questions

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 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 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 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)=\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.

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

(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.

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

(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.

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

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

(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 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.

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

(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 $

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) = \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.

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

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).

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

(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} $

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 absolute maximum and minimum values of each function

on the given interval.

b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$g(x)=e^{-x^{2}}, \quad-2 \leq x \leq 1$$

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).

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

(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) 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 absolute maximum and minimum values of each function

on the given interval.

b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$g(x)=x e^{-x}, \quad-1 \leq x \leq 1$$

(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 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.

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

a. Find the absolute maximum and minimum values of each function

on the given interval.

b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$f(x)=\frac{1}{x}+\ln x, \quad 0.5 \leq x \leq 4$$

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

$$

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.

$$f(x)=3 x^{3}+\frac{3 x^{2}}{2}-2 x \text { on } [-1,1]$$

(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.

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

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.

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

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 $f$ on the given interval.

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

c. Use a graphing utility to confirm your conclusions.

$$f(x)=x^{1 / 3}(x+4) ;[-27,27]$$

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.

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

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 $f$.

$$f(x)=x^{3}-3 x^{2} \quad(-1 \leq x \leq 3)$$

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.

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

a. Find the absolute maximum and minimum values of each functionb. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.

on the given interval.

$$h(x)=\ln (x+1)-\frac{x}{2}, \quad 0 \leq x \leq 3$$

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.

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

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?

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

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.

$$f(x)=-x^{3}+9 x ;[-4,3]$$

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?

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

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 $f$.

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

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.

$$f(x)=1 / x-\ln x$$

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 and minimum values?

$$

f^{\prime}(x)=3-\frac{6}{\sqrt{x}}, \quad x \neq 0

$$

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.

$$f(x)=x \sqrt{9-x^{2}} ;[-3,3]$$

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$F(x)=-\frac{1}{x^{2}}, \quad 0.5 \leq x \leq 2$$

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.

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

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?

Identify the inflection points and local maxima and minima of the functions graphed. Identify the intervals on which the functions are concave up and concave down. (check your book to see graph)

$$y=\frac{x^{3}}{3}-\frac{x^{2}}{2}-2 x+\frac{1}{3}$$

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.

$$f(x)=\left(e^{x}+e^{-x}\right) / 2$$

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.

$$f(x)=\frac{x^{2}}{x^{2}-1} ;[-4,4]$$

Find the $x$ -value of all points where the functions defined as follows have any relative extrema. Find the value(s) of any relative extrema.

$$f(x)=2 x-\frac{500}{x}$$

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.

$$f(x)=\frac{4 x^{5}}{5}-3 x^{3}+5 \text { on } [-2,2]$$

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

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

$ f(x) = x^3 - 3x^2 - 9x + 4 $

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

$$F(x)=-\frac{1}{x}, \quad-2 \leq x \leq-1$$

(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.

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

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

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

c. Use a graphing utility to confirm your conclusions.

$$f(x)=x / \sqrt{x-4} ;[6,12]$$