Chapter 10

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(t)=e^{t^{2}-2 t} $$

A stone is thrown straight up from the roof of an \(80-\mathrm{ft}\) building. The distance (in feet) of the stone from the ground at any time \(t\) (in seconds) is given by $$ h(t)=-16 t^{2}+64 t+80 $$ When is the stone rising, and when is it falling? If the stone were to miss the building, when would it hit the ground? Sketch the graph of \(h\). Hint: The stone is on the ground when \(h(t)=0\).

A tank initially contains 10 gal of brine with \(2 \mathrm{lb}\) of salt. Brine with \(1.5 \mathrm{lb}\) of salt per gallon enters the tank at the rate of \(3 \mathrm{gal} / \mathrm{min}\), and the well-stirred mixture leaves the tank at the rate of \(4 \mathrm{gal} / \mathrm{min}\). It can be shown that the amount of salt in the tank after \(t\) min is \(x \mathrm{lb}\) where \(x=f(t)=1.5(10-t)-0.0013(10-t)^{4} \quad(0 \leq t \leq 10)\) What is the maximum amount of salt present in the tank at any time?

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=x^{2} e^{x} $$

The Mexican subsidiary of ThermoMaster manufactures an indoor-outdoor thermometer. Management estimates that the profit (in dollars) realizable by the company for the manufacture and sale of \(x\) units of thermometers each week is $$ P(x)=-0.001 x^{2}+8 x-5000 $$ Find the intervals where the profit function \(P\) is increasing and the intervals where \(P\) is decreasing.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) is defined on a closed interval \([a, b]\), then \(f\) has an absolute maximum value.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=\ln \left(x^{2}+1\right) $$

Based on a study conducted in 1997 , the percent of the U.S. population by age afflicted with Alzheimer's disease is given by the function \(P(x)=0.0726 x^{2}+0.7902 x+4.9623 \quad(0 \leq x \leq 25)\) where \(x\) is measured in years, with \(x=0\) corresponding to age 65 yr. Show that \(P\) is an increasing function of \(x\) on the interval \((0,25)\). What does your result tell you about the relationship between Alzheimer's disease and age for the population that is age \(65 \mathrm{yr}\) and older?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) is continuous on an open interval \((a, b)\), then \(f\) does not have an absolute minimum value.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(x)=x-\ln x $$

Almost half of companies let other firms manage some of their Web operations-a practice called Web hosting. Managed services -monitoring a customer's technology services-is the fastest growing part of Web hosting. Managed services sales are expected to grow in accordance with the function $$ f(t)=0.469 t^{2}+0.758 t+0.44 \quad(0 \leq t \leq 6) $$ where \(f(t)\) is measured in billions of dollars and \(t\) is measured in years, with \(t=0\) corresponding to 1999 . a. Find the interval where \(f\) is increasing and the interval where \(f\) is decreasing. b. What does your result tell you about sales in managed services from 1999 through 2005 ?

Sketch the graph of a function having the given properties. $$ f(2)=4, f^{\prime}(2)=0, f^{\prime \prime}(x)<0 \text { on }(-\infty, \infty) $$

The height (in feet) attained by a rocket \(t\) sec into flight is given by the function $$ h(t)=-\frac{1}{3} t^{3}+16 t^{2}+33 t+10 \quad(t \geq 0) $$ When is the rocket rising, and when is it descending?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f^{\prime \prime}(x)<0\) on \((a, b)\) and \(f^{\prime}(c)=0\) where \(a

Sketch the graph of a function having the given properties. $$ \begin{array}{l} f(2)=2, f^{\prime}(2)=0, f^{\prime}(x)>0 \text { on }(-\infty, 2), f^{\prime}(x)>0 \text { on } \\ (2, \infty), f^{\prime \prime}(x)<0 \text { on }(-\infty, 2), f^{\prime \prime}(x)>0 \text { on }(2, \infty) \end{array} $$

Following the lead of the National Wildlife Federation, the Department of the Interior of a South American country began to record an index of environmental quality that measured progress and decline in the environmental quality of its forests. The index for the years 1998 through 2008 is approximated by the function $$ I(t)=\frac{1}{3} t^{3}-\frac{5}{2} t^{2}+80 \quad(0 \leq t \leq 10) $$ where \(t=0\) corresponds to 1998 . Find the intervals where the function \(I\) is increasing and the intervals where it is decreasing. Interpret your results.

Sketch the graph of a function having the given properties. $$ \begin{array}{l} f(-2)=4, f(3)=-2, f^{\prime}(-2)=0, f^{\prime}(3)=0, f^{\prime}(x)>0 \text { on } \\ (-\infty,-2) \cup(3, \infty), f^{\prime}(x)<0 \text { on }(-2,3) \text { , inflection point } \\ \text { at }(1,1) \end{array} $$

The average speed of a vehicle on a stretch of Route 134 between 6 a.m. and 10 a.m. on a typical weekday is approximated by the function $$ f(t)=20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4) $$ where \(f(t)\) is measured in miles per hour and \(t\) is measured in hours, with \(t=0\) corresponding to 6 a.m. Find the interval where \(f\) is increasing and the interval where \(f\) is decreasing and interpret your results.

Show that a polynomial function defined on the interval \((-\infty, \infty)\) cannot have both an absolute maximum and an absolute minimum unless it is a constant function.

Sketch the graph of a function having the given properties. $$ f(0)=0, f^{\prime}(0) \text { does not exist, } f^{\prime \prime}(x)<0 \text { if } x \neq 0 $$

The average cost (in dollars) incurred by Lincoln Records each week in pressing \(x\) compact discs is given by $$ \bar{C}(x)=-0.0001 x+2+\frac{2000}{x} \quad(0

One condition that must be satisfied before Theorem 3 (page 713 ) is applicable is that the function \(f\) must be continuous on the closed interval \([a, b]\). Define a function \(f\) on the closed interval \([-1,1]\) by $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{x} & \text { if } x \in[-1,1] \quad(x \neq 0) \\ 0 & \text { if } x=0 \end{array}\right. $$ a. Show that \(f\) is not continuous at \(x=0\). b. Show that \(f(x)\) does not attain an absolute maximum or an absolute minimum on the interval \([-1,1]\). c. Confirm your results by sketching the function \(f\).

Sketch the graph of a function having the given properties. $$ \begin{array}{l} f(0)=1, f^{\prime}(0)=0, f(x)>0 \text { on }(-\infty, \infty), f^{\prime \prime}(x)<0 \text { on } \\ (-\sqrt{2} / 2, \sqrt{2} / 2), f^{\prime \prime}(x)>0 \text { on }(-\infty,-\sqrt{2} / 2) \cup(\sqrt{2} / 2, \infty) \end{array} $$

Refer to Exercise 80 . Sales in the Webhosting industry are projected to grow in accordance with the function \(f(t)=-0.05 t^{3}+0.56 t^{2}+5.47 t+7.5 \quad(0 \leq t \leq 6)\) where \(f(t)\) is measured in billions of dollars and \(t\) is measured in years, with \(t=0\) corresponding to 1999 . a. Find the interval where \(f\) is increasing and the interval where \(f\) is decreasing. Hint: Use the quadratic formula. b. What does your result tell you about sales in the Webhosting industry from 1999 through 2005 ?

One condition that must be satisfied before Theorem 3 (page 713 ) is applicable is that the interval on which \(f\) is defined must be a closed interval \([a, b] .\) Define a function \(f\) on the open interval \((-1,1)\) by \(f(x)=x\). Show that \(f\) does not attain an absolute maximum or an absolute minimum on the interval \((-1,1)\). Hint: What happens to \(f(x)\) if \(x\) is close to but not equal to \(x=\) \(-1 ?\) If \(x\) is close to but not equal to \(x=1 ?\)

Sketch the graph of a function having the given properties. $$ \begin{array}{l} f \text { has domain }[-1,1], f(-1)=-1, f\left(-\frac{1}{2}\right)=-2, f^{\prime}\left(-\frac{1}{2}\right)=0, \\ f^{\prime \prime}(x)>0 \text { on }(-1,1) \end{array} $$

According to a study from the American Medical Association, the number of medical school applicants from academic year \(1997-1998(t=0)\) through the academic year 2002-2003 is approximated by the function \(N(t)=-0.0333 t^{3}+0.47 t^{2}-3.8 t+47 \quad(0 \leq t \leq 5)\) where \(N(t)\) measured in thousands. a. Show that the number of medical school applicants had been declining over the period in question. Hint: Use the quadratic formula. b. What was the largest number of medical school applicants in any one academic year for the period in question? In what academic year did that occur?

The sales of functional food products-those that promise benefits beyond basic nutrition-have risen sharply in recent years. The sales (in billions of dollars) of foods and beverages with herbal and other additives is approximated by the function \(S(t)=0.46 t^{3}-2.22 t^{2}+6.21 t+17.25 \quad(0 \leq t \leq 4)\) where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1997 . Show that \(S\) is increasing on the interval \([0,4]\). Hint: Use the quadratic formula.

Based on data from the Central Provident Fund of a certain country (a government agency similar to the Social Security Administration), the estimated cash in the fund in 2003 is given by $$ \begin{aligned} A(t)=&-96.6 t^{4}+403.6 t^{3} \\ &+660.9 t^{2}+250 \quad(0 \leq t \leq 5) \end{aligned} $$ where \(A(t)\) is measured in billions of dollars and \(t\) is measured in decades, with \(t=0\) corresponding to \(2003 .\) Find the interval where \(A\) is increasing and the interval where \(A\) is decreasing and interpret your results. Hint: Use the quadratic formula.

According to the South Coast Air Quality Management District, the level of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in downtown Los Angeles is approximated by $$ A(t)=0.03 t^{3}(t-7)^{4}+60.2 \quad(0 \leq t \leq 7) $$ where \(A(t)\) is measured in pollutant standard index (PSI) and \(t\) is measured in hours, with \(t=0\) corresponding to 7 a.m. At what time of day is the air pollution increasing, and at what time is it decreasing?

The average state cigarette tax per pack (in dollars) from 2001 through 2007 is approximated by the function $$ T(t)=0.43 t^{0.43} \quad(1 \leq t \leq 7) $$ where \(t\) is measured in years, with \(t=1\) corresponding to the beginning of 2001 . a. Show that the average state cigarette tax per pack was increasing throughout the period in question. b. What can you say about the rate at which the average state cigarette tax per pack was increasing over the period in question?

The concentration (in milligrams/cubic centimeter) of a certain drug in a patient's body \(t\) hr after injection is given by $$ C(t)=\frac{t^{2}}{2 t^{3}+1} \quad(0 \leq t \leq 4) $$ When is the concentration of the drug increasing, and when is it decreasing?

The increase in carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) in the atmosphere is a major cause of global warming. Using data obtained by Charles David Keeling, professor at Scripps Institution of Oceanography, the average amount of \(\mathrm{CO}_{2}\) in the atmosphere from 1958 through 2007 is approximated by \(A(t)=0.010716 t^{2}+0.8212 t+313.4 \quad(1 \leq t \leq 50)\) where \(A(t)\) is measured in parts per million volume (ppmv) and \(t\) in years, with \(t=1\) corresponding to the beginning of \(1958 .\) a. What can you say about the rate of change of the average amount of atmospheric \(\mathrm{CO}_{2}\) from the beginning of 1958 through 2007? b. What can you say about the rate of the rate of change of the average amount of atmospheric \(\mathrm{CO}_{2}\) from the beginning of 1958 through 2007 ?

The number of crash fatalities per 100,000 vehicle miles of travel (based on 1994 data) is approximated by the model $$ f(x)=\frac{15}{0.08333 x^{2}+1.91667 x+1} \quad(0 \leq x \leq 11) $$ where \(x\) is the age of the driver in years, with \(x=0\) corresponding to age 16 . Show that \(f\) is decreasing on \((0,11)\) and interpret your result.

The sales (in billions of dollars) in restaurants and bars in California from the beginning of \(1993(t=0)\) through \(2000(t=7)\) are approximated by the function $$ S(t)=0.195 t^{2}+0.32 t+23.7 \quad(0 \leq t \leq 7) $$ a. Show that the sales in restaurants and bars continued to rise after smoking bans were implemented in restaurants in 1995 and in bars in 1998 . Hint: Show that \(S\) is increasing in the interval \((2,7)\). b. What can you say about the rate at which the sales were rising after smoking bans were implemented?

The amount of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in the city of Long Beach is approximated by $$ A(t)=\frac{136}{1+0.25(t-4.5)^{2}}+28 \quad(0 \leq t \leq 11) $$ where \(A(t)\) is measured in pollutant standard index (PSI) and \(t\) is measured in hours, with \(t=0\) corresponding to 7 a.m. Find the intervals where \(A\) is increasing and where \(A\) is decreasing and interpret your results.

Since their introduction into the market in the late \(1990 \mathrm{~s}\), the sales of digital televisions, including high-definition television sets, have slowly gathered momentum. The model $$ S(t)=0.164 t^{2}+0.85 t+0.3 \quad(0 \leq t \leq 4) $$ describes the sales of digital television sets (in billions of dollars) between the beginning of \(1999(t=0)\) and the beginning of \(2003(t=4)\). a. Find \(S^{\prime}(t)\) and \(S^{\prime \prime}(t)\). b. Use the results of part (a) to conclude that the sales of digital TVs were increasing between 1999 and 2003 and that the sales were increasing at an increasing rate over that time interval.

The 1980s saw a trend toward oldfashioned punitive deterrence as opposed to the more liberal penal policies and community-based corrections popular in the 1960 s and early \(1970 \mathrm{~s}\). As a result, prisons became more crowded, and the gap between the number of people in prison and the prison capacity widened. The number of prisoners (in thousands) in federal and state prisons is approximated by the function $$ N(t)=3.5 t^{2}+26.7 t+436.2 \quad(0 \leq t \leq 10) $$ where \(t\) is measured in years, with \(t=0\) corresponding to 1984\. The number of inmates for which prisons were designed is given by $$ C(t)=24.3 t+365 \quad(0 \leq t \leq 10) $$ where \(C(t)\) is measured in thousands and \(t\) has the same meaning as before. Show that the gap between the number of prisoners and the number for which the prisons were designed has been widening at any time \(t\). Hint: First, write a function \(G\) that gives the gap between the number of prisoners and the number for which the prisons were designed at any time \(t\). Then show that \(G^{\prime}(t)>0\) for all values of t in the interval \((0,10)\).

An efficiency study conducted for Elektra Electronics showed that the number of Space Commander walkie-talkies assembled by the average worker \(t\) hr after starting work at 8 a.m. is given by $$ N(t)=-t^{3}+6 t^{2}+15 t \quad(0 \leq t \leq 4) $$ At what time during the morning shift is the average worker performing at peak efficiency?

U.S. NURSING SHORTAGE The demand for nurses between 2000 and 2015 is estimated to be $$ D(t)=0.0007 t^{2}+0.0265 t+2 \quad(0 \leq t \leq 15) $$ where \(D(t)\) is measured in millions and \(t=0\) corresponds to the year 2000 . The supply of nurses over the same time period is estimated to be $$ S(t)=-0.0014 t^{2}+0.0326 t+1.9 \quad(0 \leq t \leq 15) $$ where \(S(t)\) is also measured in millions. a. Find an expression \(G(t)\) giving the gap between the demand and supply of nurses over the period in question. b. Find the interval where \(G\) is decreasing and where it is increasing. Interpret your result. c. Find the relative extrema of \(G\). Interpret your result.

The altitude (in feet) of a rocket \(t\) sec into flight is given by $$ s=f(t)=-t^{3}+54 t^{2}+480 t+6 \quad(t \geq 0) $$ Find the point of inflection of the function \(f\) and interpret your result. What is the maximum velocity attained by the rocket?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) is decreasing on \((a, b)\), then \(f^{\prime}(x)<0\) for each \(x\) in \((a, b) .\)

In a study conducted in 2003 , business spending on technology (in billions of dollars) from the beginning of 2000 through 2005 was projected to be \(S(t)=-1.88 t^{3}+30.33 t^{2}-76.14 t+474 \quad(0 \leq t \leq 5)\) where \(t\) is measured in years, with \(t=0\) corresponding to 2000\. Show that the graph of \(S\) is concave upward on the interval \((0,5)\). What does this result tell you about the rate of business spending on technology over the period in question?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) and \(g\) are hoth increasing on \((a, b)\), then \(f+g\) is increasing on \((a, b)\).

Congress created the alternative minimum tax (AMT) in the late 1970 s to ensure that wealthy people paid their fair share of taxes. But because of quirks in the law, even middle-income taxpayers have started to get hit with the tax. The AMT (in billions of dollars) projected to be collected by the IRS from the beginning of 2001 through 2010 is \(f(t)=0.0117 t^{3}+0.0037 t^{2}+0.7563 t+4.1 \quad(0 \leq t \leq 9)\) where \(t\) is measured in years, with \(t=0\) corresponding to \(2001 .\) a. Show that \(f\) is increasing on the interval \((0,9)\). What does this result tell you about the projected amount of AMT paid over the years in question? b. Show that \(f^{\prime}\) is increasing on the interval \((0,9)\). What conclusion can you draw from this result concerning the rate of growth at which the AMT is paid over the years in question?

The total annual revenue \(R\) of the Miramar Resorts Hotel is related to the amount of money \(x\) the hotel spends on advertising its services by the function \(R(x)=-0.003 x^{3}+1.35 x^{2}+2 x+8000 \quad(0 \leq x \leq 400)\) where both \(R\) and \(x\) are measured in thousands of dollars. a. Find the interval where the graph of \(R\) is concave upward and the interval where the graph of \(R\) is concave downward. What is the inflection point of \(R\) ? b. Would it be more beneficial for the hotel to increase its advertising budget slightly when the budget is $$\$ 140,000$$ or when it is $$\$ 160,000 ?$$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f(x)\) and \(g(x)\) are positive on \((a, b)\) and both \(f\) and \(g\) are increasing on \((a, b)\), then \(f g\) is increasing on \((a, b)\).

As a result of increasing energy costs, the growth rate of the profit of the 4-yr old Venice Glassblowing Company has begun to decline. Venice's management, after consulting with energy experts, decides to implement certain energy-conservation measures aimed at cutting energy bills. The general manager reports that, according to his calculations, the growth rate of Venice's profit should be on the increase again within 4 yr. If Venice's profit (in hundreds of dollars) \(t\) yr from now is given by the function $$ P(t)=t^{3}-9 t^{2}+40 t+50 \quad(0 \leq t \leq 8) $$ determine whether the general manager's forecast will be accurate. Hint: Find the inflection point of the function \(P\) and study the concavity of \(P\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f^{\prime}(c)=0\), then \(f\) has a relative maximum or a relative minimum at \(x=c\).

The amount (in billions of dollars) spent by the top 15 U.S. financial institutions on IT (information technology) offshore outsourcing is projected to be $$ A(t)=0.92(t+1)^{0.61} \quad(0 \leq t \leq 4) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 2004 . a. Show that \(A\) is increasing on \((0,4)\) and interpret your result. b. Show that \(A\) is concave downward on \((0,4)\). Interpret your result.