## Educators

MP
FP
WZ

### Problem 1

Explain the difference between an absolute minimum and a local minimum.

FP

### Problem 2

Suppose $f$ is a continuous function defined on a closed interval $[a, b]$.

(a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for $f$?
(b) What steps would you take to find those maximum and minimum values?

FP

### Problem 3

For each of the numbers $a$, $b$, $c$, $d$, $r$, and $s$, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

FP

### Problem 4

For each of the numbers $a$, $b$, $c$, $d$, $r$, and $s$, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

WZ
Wen Z.

### Problem 5

Use the graph to state the absolute and local maximum and minimum values of the function.

FP

### Problem 6

Use the graph to state the absolute and local maximum and minimum values of the function.

WZ
Wen Z.

### Problem 7

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 $4$

FP

### Problem 8

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 $3$

FP

### Problem 9

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 $2$

FP

### Problem 10

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

Absolute maximum at $2$, absolute minimum at $5$, $4$ is a critical number but there is no local maximum and minimum there.

FP

### Problem 11

(a) Sketch the graph of a function that has a local maximum of $2$ and is differentiable at $2$.
(b) Sketch the graph of a function that has a local maximum of $2$ and is continuous but not differentiable at $2$.
(c) Sketch the graph of a function that has a local maximum of $2$ and is not continuous at $2$.

FP

### Problem 12

(a) Sketch the graph of a function on $[-1, 2]$ that has an absolute maximum but no local maximum.
(b) Sketch the graph of a function on $[-1, 2]$ that has a local maximum but no absolute maximum.

FP

### Problem 13

(a) Sketch the graph of a function on $[-1, 2]$ that has an absolute maximum but no absolute minimum.
(b) Sketch the graph of a function on $[-1, 2]$ that is discontinuous but has both an absolute maximum and an absolute minimum.

FP

### Problem 14

(a) Sketch the graph of a function that has two local maximum, one local minimum, and no absolute minimum.
(b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

FP

### Problem 15

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).

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

FP

### Problem 16

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).

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

FP

### Problem 17

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).

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

FP

### Problem 18

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).

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

FP

### Problem 19

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).

$f(x) = \sin x$, $0 \leqslant x < \pi /2$

FP

### Problem 20

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).

$f(x) = \sin x$, $0 < x \leqslant \pi /2$

FP

### Problem 21

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 tranformations of Section 1.2 and 1.3).

$f(x) = \sin x$, $-\pi /2 \leqslant x \leqslant \pi /2$

FP

### Problem 22

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).

$f(x) = \cos t$, $-3 \pi /2 \leqslant t \leqslant 3\pi /2$

FP

### Problem 23

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).

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

FP

### Problem 24

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).

$f(x) = | x |$

FP

### Problem 25

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).

$f(x) = 1 - \sqrt{x}$

FP

### Problem 26

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).

$f(x) = e^x$

FP

### Problem 27

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).

$f(x) = \left\{ \begin{array}{ll} x^2 & \mbox{ if}-1 \leqslant x \leqslant 0\\ 2 - 3x & \mbox{ if} 0 < x \leqslant 1 \end{array} \right.$

FP

### Problem 28

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).

$f(x) = \left\{ \begin{array}{ll} 2x + 1 & \mbox{ if} 0 \leqslant x < 1\\ 4 - 2x & \mbox{ if} 1 \leqslant x \leqslant 3 \end{array} \right.$

FP

### Problem 29

Find the critical numbers of the function.

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

FP

### Problem 30

Find the critical numbers of the function.

$f(x) = x^3 + 6x^2 - 15x$

### Problem 31

Find the critical numbers of the function.

$f(x) = 2x^3 - 3x^2 - 36x$

Amrita B.

### Problem 32

Find the critical numbers of the function.

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

Amrita B.

### Problem 33

Find the critical numbers of the function.

$g(t) = t^4 + t^3 + t^2 + 1$

Amrita B.

### Problem 34

Find the critical numbers of the function.

$g(t) = | 3t - 4 |$

Amrita B.

### Problem 35

Find the critical numbers of the function.

$g(y) = \frac{ y - 1}{ y^2 - y + 1}$

Amrita B.

### Problem 36

Find the critical numbers of the function.

$h(p) = \frac{p - 1}{p^2 + 4}$

Amrita B.

### Problem 37

Find the critical numbers of the function

$h(t) = t^\frac{3}{4} - 2t^\frac{1}{4}$

Amrita B.

### Problem 38

Find the critical numbers of the function.

$g(x) = \sqrt[3]{4 - x^2}$

Amrita B.

### Problem 39

Find the critical numbers of the function.

$F(x) = x^\frac{4}{5} (x - 4)^2$

Amrita B.

### Problem 40

Find the critical numbers of the function.

$g(\theta) = 4\theta - \tan\theta$

Amrita B.

### Problem 41

Find the critical numbers of the function.

$f(\theta) = 2 \cos \theta + \sin^2 \theta$

Amrita B.

### Problem 42

Find the critical numbers of the function.

$h(t) = 3t - \arcsin t$

Amrita B.

### Problem 43

Find the critical numbers of the function.

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

Amrita B.

### Problem 44

Find the critical numbers of the function.

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

Amrita B.

### Problem 45

A formula for the derivative of a function $f$ is given. How many critical numbers does $f$ have?

$f'(x) = 5e^{-0.1 |x|} \sin x - 1$

Amrita B.

### Problem 46

A formula for the derivative of a function $f$ is given. How many critical numbers does $f$ have?

$f'(x) = \frac{100 \cos^2 x}{10 + x^2} - 1$

Amrita B.

### Problem 47

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

$f(x) = 12 + 4x - x^2$, $[0, 5]$

Amrita B.

### Problem 48

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

$f(x) = 5 + 54x - 2x^3$, $[0, 4]$

Amrita B.

### Problem 49

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

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

Amrita B.

### Problem 50

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

$f(x) = x^3 - 6x^2 + 5$, $[-3, 5]$

Amrita B.

### Problem 51

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

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

Amrita B.

### Problem 52

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

$f(t) = (t^2 - 4)^3$, $[-2, 3]$

Amrita B.

### Problem 53

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

$f(x) = x + \frac{1}{x}$, $[0.2, 4]$

Amrita B.

### Problem 54

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

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

Amrita B.

### Problem 55

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

$f(t) = t - \sqrt[3]{x}$, $[-1, 4]$

Amrita B.

### Problem 56

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

$f(t) = \frac{\sqrt{t}}{1 + t^2}$, $[0, 2]$

Amrita B.

### Problem 57

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

$f(t) = 2 \cos t + \sin 2t$, $[0, \pi /2]$

Amrita B.

### Problem 58

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

$f(t) = t + \cot (t/2)$, $[\pi /4, 7\pi /4]$

Amrita B.

### Problem 59

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

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

Amrita B.

### Problem 60

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

$f(x) = xe^{x/2}$, $[-3, 1]$

Amrita B.

### Problem 61

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

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

Amrita B.

### Problem 62

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

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

Amrita B.

### Problem 63

If $a$ and $b$ are positive numbers, find the maximum value of $f(x) = x^a (1 - x)^b$, $0 \leqslant x \leqslant 1$.

Amrita B.

### Problem 64

Use a graph to estimate the critical numbers of $f(x) = | 1 + 5x - x^3 |$ correct to one decimal place.

Amrita B.

### Problem 65

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

$f(x) = x^5 - x^3 + 2$, $-1 \leqslant x \leqslant 1$

Amrita B.

### Problem 66

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

$f(x) = e^x + e^{-2x}$, $0 \leqslant x \leqslant 1$

Amrita B.

### Problem 67

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

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

Amrita B.

### Problem 68

(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum values.

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

Amrita B.

### Problem 69

After the consumption of an alcoholic beverage, the concentration of alcohol in the bloodstream (blood alcohol concentration, or BAC) surges as the alcohol is absorbed, followed by a gradual decline as the alcohol is metabolized. The function

$$C(t) = 1.35te^{-2.802t}$$

models the average BAC, measured in mg/mL, in a group of eight male subjects $t$ hours after rapid consumption of $15$ ml of ethanol (corresponding to one alcoholic drink). What is the maximum average BAC during the first $3$ hours? When does it occur?

Source: Adapted from P. Wilkinson et. al., "Pharmacokinetics of Ethanol after Oral Administration in the Fasting State," Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207-24.

Amrita B.

### Problem 70

After an antibiotic taken is taken, the concentration of the antibiotic in the bloodstream is modeled by the function

$$C(t) = 8(e^{-0.4t} - e^{-0.6t})$$

where the time $t$ is measured in hours and $C$ is measured in $\mu$g/mL. What is the maximum concentration of the antibiotic during the first $12$ hours?

Amrita B.

### Problem 71

Between 0 $^\circ C$ and 30 $^\circ C$, the volume $V$ (in cubic centimeters) of $1$ kg of water at a temperature $T$ is given approximately by the formula
$$V = 999.87 - 0.06426T + 0.0085043T^3 - 0.0000679T^3$$
Find the temperature at which water has its maximum density.

Amrita B.

### Problem 72

An object with weight $W$ is dragged along a horizontal plane by force acting along a rope attached to the object. If the rope makes an angle $\theta$ with the plane, then the magnitude of the force is
$$F = \frac{\mu W}{\mu \sin \theta + \cos \theta}$$
where $\mu$ is a positive constant called the coefficient of friction and where $0 \leqslant \theta \leqslant \pi /2$. Show that $F$ is minimized when $\tan \theta = \mu$.

Amrita B.

### Problem 73

The water level, measured in feet above mean sea level, of Lake Lanier in Georgia, USA, during 2012 can be modeled by the function
$$L(t) = 0.01441t^3 - 0.4177t^3 + 2.703t + 1060.1$$
where $t$ is measured in months since January 1, 2012. Estimate when the water level was highest during 2012.

Amrita B.

### Problem 74

On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters.
(a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval $t \in [0, 125]$. Then graph this polynomial.
(b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first $125$ seconds.

Amrita B.

### Problem 75

When a foreign object lodged in the trachea (wind pipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction in the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the air-stream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity $v$ of the air-stream is related to the radius $r$ of the trachea by the equation
$$v(r) = k(r_o - r) r^2$$ $$\frac{1}{2}r_o \leqslant r \leqslant r_o$$
where $k$ is constant and $r_o$ is the normal radius of the trachea.
The restriction of $r$ is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than $\frac{1}{2}r_o$ is prevented (otherwise the person would suffocate.
(a) Determine the value of $r$ in the interval $[\frac{1}{2}r_o, r_o]$ at which $v$ has an absolute maximum. How does this compare with experimental evidence?
(b) What is the absolute maximum value of $v$ on the interval?
(c) Sketch the graph of $v$ on the interval $[0, r_o]$.

WZ
Wen Z.

### Problem 76

Show that $5$ is a critical number of the function
$$g(x) = 2 + (x - 5)^3$$
but $g$ does not have a local extreme value at $5$.

Amrita B.

### Problem 77

Prove that the function
$$f(x) = x^101 + x^51 + x + 1$$
has neither a local maximum nor a local minimum.

Amrita B.

### Problem 78

If $f$ has a local minimum value at $c$, show that the function $g(x) = - f(x)$ has a local maximum value at $c$.

Amrita B.

### Problem 79

Prove Fermat's Theorem for the case in which $f$ has a local minimum at $c$.

Amrita B.
A cubic function is a polynomial of degree $3$; that is, it has the form $f(x) = ax^3 + bx^2 + cx + d$, where $a \not= 0$.