Chapter 10

Using Theorem 1 , verify that the linear function \(f(x)=m x+\) \(b\) is (a) increasing everywhere if \(m>0\), (b) decreasing everywhere if \(m<0\), and \((\mathrm{c})\) constant if \(m=0\).

Medicaid spending on drugs in Massachusetts started slowing down in part after the state demanded that patients use more generic drugs and limited the range of drugs available to the program. The annual pharmacy spending (in millions of dollars) from 1999 through 2004 is given by \(S(t)=-1.806 t^{3}+10.238 t^{2}+93.35 t+583 \quad(0 \leq t \leq 5)\) where \(t\) is measured in years with \(t=0\) corresponding to the beginning of \(1999 .\) Find the inflection point of \(S\) and interpret your result.

Show that the function \(f(x)=x^{3}+x+1\) has no relative extrema on \((-\infty, \infty)\).

Research reports indicate that surveillance cameras at major intersections dramatically reduce the number of drivers who barrel through red lights. The cameras automatically photograph vehicles that drive into intersections after the light turns red. Vehicle owners are then mailed citations instructing them to pay a fine or sign an affidavit that they weren't driving at the time. The function \(N(t)=6.08 t^{3}-26.79 t^{2}+53.06 t+69.5 \quad(0 \leq t \leq 4)\) gives the number, \(N(t)\), of U.S. communities using surveillance cameras at intersections in year \(t\), with \(t=0\) corresponding to the beginning of 2003 . a. Show that \(N\) is increasing on \([0,4]\). b. When was the number of communities using surveillance cameras at intersections increasing least rapidly? What is the rate of increase?

Let \(f(x)=x^{2}+a x+b\). Determine the constants \(a\) and \(b\) so that \(f\) has a relative minimum at \(x=2\) and the relative minimum value is 7 .

The revenue for Google from the beginning of \(1999(t=0)\) through \(2003(t=4)\) is approximated by the function \(R(t)=24.975 t^{3}-49.81 t^{2}+41.25 t+0.2 \quad(0 \leq t \leq 4)\) where \(R(t)\) is measured in millions of dollars. a. Find \(R^{\prime}(t)\) and \(R^{\prime \prime}(t)\). b. Show that \(R^{\prime}(t)>0\) for all \(t\) in the interval \((0,4)\) and interpret your result. Hint: Use the quadratic formula. c. Find the inflection point of \(R\) and interpret your result.

Clark County in Nevada-dominated by greater Las Vegas-is one of the fastest- growing metropolitan areas in the United States. The population of the county from 1970 through 2000 is approximated by the function $$ \begin{array}{r} P(t)=44560 t^{3}-89394 t^{2}+234633 t+273288 \\ (0 \leq t \leq 4) \end{array} $$ where \(t\) is measured in decades, with \(t=0\) corresponding to the beginning of 1970 . a. Show that the population of Clark County was always increasing over the time period in question. Hint: Show that \(P^{\prime}(t)>0\) for all \(t\) in the interval \((0,4)\). b. Show that the population of Clark County was increasing at the slowest pace some time toward the middle of August 1976 . Hint: Find the inflection point of \(P\) in the interval \((0,4)\).

Let $$ f(x)=\left\\{\begin{array}{ll} -3 x & \text { if } x<0 \\ 2 x+4 & \text { if } x \geq 0 \end{array}\right. $$ a. Compute \(f^{\prime}(x)\) and show that it changes sign from negative to positive as we move across \(x=0\). b. Show that \(f\) does not have a relative minimum at \(x=0 .\) Does this contradict the first derivative test? Explain your answer.

Measles is still a leading cause of vaccine-preventable death among children, but due to improvements in immunizations, measles deaths have dropped globally. The function \(N(t)=-2.42 t^{3}+24.5 t^{2}-123.3 t+506 \quad(0 \leq t \leq 6)\) gives the number of measles deaths (in thousands) in subSaharan Africa in year \(t\), with \(t=0\) corresponding to \(1999 .\) a. How many measles deaths were there in \(1999 ?\) In \(2005 ?\) b. Show that \(N^{\prime}(t)<0\) on \((0,6)\). What does this say about the number of measles deaths from 1999 through 2005 ? c. When was the number of measles deaths decreasing most rapidly? What was the rate of measles death at that instant of time?

Let $$ f(x)=\left\\{\begin{array}{ll} -x^{2}+3 & \text { if } x \neq 0 \\ 2 & \text { if } x=0 \end{array}\right. $$ a. Compute \(f^{\prime}(x)\) and show that it changes sign from positive to negative as we move across \(x=0\). b. Show that \(f\) does not have a relative maximum at \(x=0\). Does this contradict the first derivative test? Explain your answer.

Many public entities like cities, counties, states, utilities, and Indian tribes are hiring firms to lobby Congress. One goal of such lobbying is to place earmarks-money directed at a specific project-into appropriation bills. The amount (in millions of dollars) spent by public entities on lobbying from 1998 through 2004 is given by $$ \begin{array}{r} f(t)=-0.425 t^{3}+3.6571 t^{2}+4.018 t+43.7 \\ (0 \leq t \leq 6) \end{array} $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1998 . a. Show that \(f\) is increasing on \((0,6)\). What does this say about the spending by public entities on lobbying over the years in question? b. Find the inflection point of \(f\). What does your result tell you about the growth of spending by the public entities on lobbying?

Let $$ f(x)=\left\\{\begin{array}{ll} \frac{1}{x^{2}} & \text { if } x>0 \\ x^{2} & \text { if } x \leq 0 \end{array}\right. $$ a. Compute \(f^{\prime}(x)\) and show that it does not change sign as we move across \(x=0\). b. Show that \(f\) has a relative minimum at \(x=0\). Does this contradict the first derivative test? Explain your answer.

The level of ozone, an invisible gas that irritates and impairs breathing, present in the atmosphere on a certain May day in the city of Riverside was approximated by $$ A(t)=1.0974 t^{3}-0.0915 t^{4} \quad(0 \leq t \leq 11) $$ where \(A(t)\) is measured in pollutant standard index (PSI) and \(t\) is measured in hours, with \(t=0\) corresponding to 7 a.m. Use the second derivative test to show that the function \(A\) has a relative maximum at approximately \(t=9\). Interpret your results.

Show that the quadratic function $$ f(x)=a x^{2}+b x+c \quad(a \neq 0) $$ has a relative extremum when \(x=-b / 2 a\). Also, show that the relative extremum is a relative maximum if \(a<0\) and a relative minimum if \(a>0\).

Based on company financial reports, the cash reserves of Blue Cross and Blue Shield as of the beginning of year \(t\) is approximated by the function $$ \begin{array}{r} R(t)=-1.5 t^{4}+14 t^{3}-25.4 t^{2}+64 t+290 \\ (0 \leq t \leq 6) \end{array} $$ where \(R(t)\) is measured in millions of dollars and \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1998 . a. Find the inflection points of \(R\). Hint: Use the quadratic formula. b. Use the result of part (a) to show that the cash reserves of the company was growing at the greatest rate at the beginning of 2002 .

Show that the cubic function $$ f(x)=a x^{3}+b x^{2}+c x+d \quad(a \neq 0) $$ has no relative extremum if and only if \(b^{2}-3 a c \leq 0\).

WoMEN's Soccer Starting with the youth movement that took hold in the \(1970 \mathrm{~s}\) and buoyed by the success of the U.S. national women's team in international competition in recent years, girls and women have taken to soccer in ever-growing numbers. The function $$ \begin{array}{r} N(t)=-0.9307 t^{3}+74.04 t^{2}+46.8667 t+3967 \\ (0 \leq t \leq 16) \end{array} $$ gives the number of participants in women's soccer in year \(t\), with \(t=0\) corresponding to the beginning of 1985 . a. Verify that the number of participants in women's soccer had been increasing from 1985 through 2000 . Hint: Use the quadratic formula. b. Show that the number of participants in women's soccer had been increasing at an increasing rate from 1985 through 2000 . Hint: Show that the sign of \(N^{\prime \prime}\) is positive on the interval in question.

Refer to Example 6, page 561 . a. Show that \(f\) is increasing on the interval \((0,1)\). b. Show that \(f(0)=-1\) and \(f(1)=1\) and use the result of part (a) together with the intermediate value theorem to conclude that there is exactly one root of \(f(x)=\) 0 in \((0,1)\)

Show that the function $$ f(x)=\frac{a x+b}{c x+d} $$ does not have a relative extremum if \(a d-b c \neq 0 .\) What can you say about \(f\) if \(a d-b c=0\) ?

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 the graph of \(f\) is concave upward on \((a, b)\), then the graph of \(-f\) is concave downward on \((a, b)\).

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 the graph of \(f\) is concave upward on \((a, c)\) and concave downward on \((c, b)\), where \(a

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 \(c\) is a critical number of \(f\) where \(a

Show that the quadratic function $$ f(x)=a x^{2}+b x+c \quad(a \neq 0) $$ is concave upward if \(a>0\) and concave downward if \(a<0\). Thus, by examining the sign of the coefficient of \(x^{2}\), one can tell immediately whether the parabola opens upward or downward.

Consider the functions \(f(x)=x^{3}, g(x)=x^{4}\), and \(h(x)=-x^{4}\) a. Show that \(x=0\) is a critical number of each of the functions \(f, g\), and \(h .\) b. Show that the second derivative of each of the functions \(f, g\), and \(h\) equals zero at \(x=0\). c. Show that \(f\) has neither a relative maximum nor a relative minimum at \(x=0\), that \(g\) has a relative minimum at \(x=0\), and that \(h\) has a relative maximum at \(x=0\).