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 $2.3 .$ .

## Discussion

## Video Transcript

No transcript available

## Recommended Questions

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=e^{-x} \cos x, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

$$

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.

$$f(x)=x \sin ^{-1} x \text { on } [-1,1]$$

Find the critical points of $f(x)=\sin x+\cos x$ and determine the extreme values on $\left[0, \frac{\pi}{2}\right]$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=\sin \theta+\sqrt{3} \cos \theta

$$

Find the critical point of

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

in the open first quadrant $(x>0, y>0)$ and show that $f$ takes on

a minimum there.

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.

$$f(x)=2 x^{6}-15 x^{4}+24 x^{2} \text { on } [-2,2]$$

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.

$$f(x)=2^{x} \sin x ;[-2,6]$$

Find all critical numbers by hand. If available, use graphing technology to determine whether the critical number represents a local maximum, local minimum or neither.

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

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).

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

Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. (GRAPH CAN'T COPY)

Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. (GRAPH CAN'T COPY)

Indicate all critical points of the function f. How many critical points are there? Identify each critical point as a local maximum, a local minimum, or neither. (GRAPH CAN'T COPY)

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.

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

Find all critical numbers by hand. Use your knowledge of the type of graph (i.e., parabola or cubic) to determine whether the critical number represents a local maximum, local minimum or neither.

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

In Exercises $29-58,$ find the min and max of the function on the given interval by comparing values at the critical points and endpoints.

$$

y=\sin x \cos x, \quad\left[0, \frac{\pi}{2}\right]

$$

In Exercises $19-22,$ find critical points of $f$ and use the First Derivative Test to determine whether they are local minima or maxima

$$

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

$$

Absolute maxima and minima Determine the location and value of the absolute extreme values of $f$ on the given interval, if they exist.

$$f(x)=2 x^{6}-15 x^{4}+24 x^{2} \text { on }[-2,2]$$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=\frac{1}{3} x^{3}+\frac{3}{2} x^{2}+2 x+4

$$

Find all critical numbers by hand. If available, use graphing technology to determine whether the critical number represents a local maximum, local minimum or neither.

$$f(x)=\sqrt{3} \sin x+\cos x$$

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.

$$h(x)=\frac{5-x}{x^{2}+2 x-3} \text { on }[-10,10]$$

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).

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

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=\tan ^{-1} x-\frac{1}{2} x

$$

Use a graphing device as in Example 4 (or Newton's method or solve numerically using a calculator or computer) to find the critical points of $ f $ correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any.

$ f(x, y) = 20e^{-x^2 - y^2} \sin 3x \cos 3y $, $ | x | \leqslant 1 $, $ | y | \leqslant 1 $

Find any critical numbers for $f$ in Exercises $59-66$ and then use the second derivative test to decide whether the eritical numbers lead to relative maxima or relative minima. If $f^{\prime \prime}(c)=0$ or $f^{\prime \prime}(c)$ does not exist for a critical number $c,$ then the second derivative test gives no information. In this case, use the first derivative test instead.

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

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=\theta-2 \cos \theta, \quad | 0,2 \pi ]

$$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=x+e^{-x}

$$

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = x - 2 \tan^{-1} x $, $ [0, 4] $

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.

$$f(x)=\frac{4 x^{3}}{3}+5 x^{2}-6 x \text { on } [-4,1]$$

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = \frac{x}{x^2 - x + 1} $, $ [0, 3] $

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=x^{-2}-4 x^{-1} \quad(x>0)

$$

Approximate the critical points of $g(x)=x \cos ^{-1} x$ and estimate the maximum value of $g(x) .$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=\left(x^{2}-2 x\right) 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).

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

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=\left\{\begin{array}{ll}{3-x,} & {x<0} \\ {3+2 x-x^{2},} & {x \geq 0}\end{array}\right.

$$

Find the absolute maximum and absolute minimum values of $f$ on the given interval.

$ f(x) = \ln (x^2 + x + 1) $, $ [-1, 1] $

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.

$$f(x)=2 x^{3}-15 x^{2}+24 x \text { on } [0,5]$$

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.

$$f(x)=x /\left(x^{2}+1\right)^{2} \text { on } [-2,2]$$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=-x^{2}+7 x-17

$$

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.

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

Find any critical numbers for $f$ in Exercises $57-64$ and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If $f^{\prime \prime}(c)=0$ or $f^{\prime \prime}(c)$ does not exist for a critical number $c,$ then the second derivative test gives no information. In this case, use the first derivative test instead.

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

Find all critical numbers by hand. Use your knowledge of the type of graph (i.e., parabola or cubic) to determine whether the critical number represents a local maximum, local minimum or neither.

$$f(x)=x^{2}+5 x-1$$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=x^{2} e^{x}

$$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=x^{2}+(10-x)^{2}

$$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=x-\ln x \quad(x>0)

$$

Absolute maxima and minima Determine the location and value of the absolute extreme values of $f$ on the given interval, if they exist.

$$f(x)=2 x^{3}-15 x^{2}+24 x \text { on }[0,5]$$

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.

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

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=\frac{1}{x^{2}+1}

$$

In Exercises $23-52,$ find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).

$$

y=x^{5}+x^{3}+x

$$

Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible ) whether they correspond to local maxima or local minima.

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