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}. $ $$ f(x)=\sqrt{3} \cos x+\sin x, \quad 0 \leq x \leq 2 \pi $$

## Discussion

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

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.

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

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?

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

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)

$$y=2 \cos x-\sqrt{2} x,-\pi \leq x \leq \frac{3 \pi}{2}$$

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) Use a graph of $ f $ to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.

(b) use a graph of $ f" $ to give better estimates.

$ f(x) = \sin 2x + \sin 4x $, $ 0 \leqslant x \leqslant \pi $

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 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.

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

(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) = 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^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) = x^4e^{-x} 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) 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) 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.

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

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.

\begin{equation}

y=2 \cos x-\sqrt{2} x,-\pi \leq x \leq \frac{3 \pi}{2}

\end{equation}

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.

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

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.

$\begin{equation}

y=\sin |x|,-2 \pi \leq x \leq 2 \pi

\end{equation}$

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.

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

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).

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

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.

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

(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.

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

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.

$$f(x)=\sec x ;[-\pi / 4, \pi / 4]$$

(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.

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

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 $f$.

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

(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 $

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.

\begin{equation}

y=x+\sin 2 x,-\frac{2 \pi}{3} \leq x \leq \frac{2 \pi}{3}

\end{equation}

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) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

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

(d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.

$f(\theta)=2 \cos \theta+\cos ^{2} \theta, \quad 0 \leqslant \theta \leqslant 2 \pi$

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.

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

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.

$$f(x)=\sin 3 x \text { on } [-\pi / 4, \pi / 3]$$

(a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

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

(d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.

$S(x)=x-\sin x, \quad 0 \leqslant x \leqslant 4 \pi$

(a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

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

(d) Use the information from parts $ (a) - (c) $ to sketch the graph.

Check your work with a graphing device if you have one.

$ f(\theta) = 2 \cos \theta + \cos^2 \theta $, $ 0 \leqslant \theta \leqslant 2\pi $

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 $f$

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