Section 1
3.1 Exercises
Explain the difference between an absolute minimum and a local minimum.
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?
3-4 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.(GRAPH CAN'T COPY)
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.(GRAPH CAN'T COPY)
5-6 Use the graph to state the absolute and local maximum and minimum values of the function.(GRAPH CAN'T COPY)
Use the graph to state the absolute and local maximum and minimum values of the function.(GRAPH CAN'T COPY)
7-10 Sketch the graph of a function $f$ that is continuous on $[1,5]$ and has the given properties.Absolute minimum at 2 , absolute maximum at 3 , local minimum at 4
Sketch the graph of a function $f$ that is continuous on $[1,5]$ and has the given properties.Absolute minimum at 1 , absolute maximum at 5 , local maximum at 2 , local minimum at 4
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
Sketch the graph of a function $f$ that is continuous on $[1,5]$ and has the given properties.$f$ has no local maximum or minimum, but 2 and 4 are critical numbers
(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2 .(b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2 .(c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2 .
(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.
(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.
(a) Sketch the graph of a function that has two local maxima, 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.
15-28 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 Sections 1.2 and 1.3.)$f(x)=\frac{1}{2}(3 x-1), \quad x \leqslant 3$
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 Sections 1.2 and 1.3.)$f(x)=2-\frac{1}{3} x, \quad x \geqslant-2$
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 Sections 1.2 and 1.3.)$f(x)=1 / x, \quad x \geqslant 1$
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 Sections 1.2 and 1.3.)$f(x)=1 / x, \quad 1<x<3$
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 Sections 1.2 and 1.3.)$f(x)=\sin x, \quad 0 \leqslant x<\pi / 2$
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 Sections 1.2 and 1.3.)$f(x)=\sin x, \quad 0<x \leqslant \pi / 2$
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 Sections 1.2 and 1.3.)$f(x)=\sin x, \quad-\pi / 2 \leqslant x \leqslant \pi / 2$
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 Sections 1.2 and 1.3.)$f(t)=\cos t, \quad-3 \pi / 2 \leqslant t \leqslant 3 \pi / 2$
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 Sections 1.2 and 1.3.)$f(x)=1+(x+1)^2, \quad-2 \leqslant x<5$
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 Sections 1.2 and 1.3.)$f(x)=|x|$
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 Sections 1.2 and 1.3.)$f(x)=1-\sqrt{x}$
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 Sections 1.2 and 1.3.)$f(x)=1-x^3$
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 Sections 1.2 and 1.3.)$f(x)= \begin{cases}1-x & \text { if } 0 \leq x<2 \\ 2 x-4 & \text { if } 2 \leq x \leq 3\end{cases}$
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 Sections 1.2 and 1.3.)$f(x)= \begin{cases}4-x^2 & \text { if }-2 \leq x<0 \\ 2 x-1 & \text { if } 0 \leq x \leq 2\end{cases}$
29-42 Find the critical numbers of the function.$f(x)=4+\frac{1}{3} x-\frac{1}{2} x^2$
Find the critical numbers of the function.$f(x)=x^3+6 x^2-15 x$
Find the critical numbers of the function.$f(x)=2 x^3-3 x^2-36 x$
Find the critical numbers of the function.$f(x)=2 x^3+x^2+2 x$
Find the critical numbers of the function.$g(t)=t^4+t^3+t^2+1$
Find the critical numbers of the function.$g(t)=|3 t-4|$
Find the critical numbers of the function.$g(y)=\frac{y-1}{y^2-y+1}$
Find the critical numbers of the function.$h(p)=\frac{p-1}{p^2+4}$
Find the critical numbers of the function.$h(t)=t^{3 / 4}-2 t^{1 / 4}$
Find the critical numbers of the function.$g(x)=x^{1 / 3}-x^{-2 / 3}$
Find the critical numbers of the function.$F(x)=x^{4 / 5}(x-4)^2$
Find the critical numbers of the function.$g(\theta)=4 \theta-\tan \theta$
Find the critical numbers of the function.$f(\theta)=2 \cos \theta+\sin ^2 \theta$
Find the critical numbers of the function.$g(x)=\sqrt{1-x^2}$
43-44 A formula for the derivative of a function $f$ is given. How many critical numbers does $f$ have?$f^{\prime}(x)=1+\frac{210 \sin x}{x^2-6 x+10}$
A formula for the derivative of a function $f$ is given. How many critical numbers does $f$ have?$f^{\prime}(x)=\frac{100 \cos ^2 x}{10+x^2}-1$
45-56 Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)=12+4 x-x^2, \quad[0,5]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)=5+54 x-2 x^3, \quad[0,4]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)=2 x^3-3 x^2-12 x+1, \quad[-2,3]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)-x^3-6 x^2+5,[-3,5]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)=3 x^4-4 x^3-12 x^2+1, \quad[-2,3]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)=\left(x^2-1\right)^3, \quad[-1,2]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)=x+\frac{1}{x}, \quad[0.2,4]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(x)=\frac{x}{x^2-x+1}, \quad[0,3]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(t)=t \sqrt{4-t^2}, \quad[-1,2]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(t)=\sqrt[3]{t}(8-t), \quad[0,8]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(t)=2 \cos t+\sin 2 t, \quad[0, \pi / 2]$
Find the absolute maximum and absolute minimum values of $f$ on the given interval.$f(t)=t+\cot (t / 2), \quad[\pi / 4,7 \pi / 4]$
If $a$ and $b$ are positive numbers, find the maximum value of $f(x)=x^a(1-x)^b, 0 \leqslant x \leqslant 1$.
Use a graph to estimate the critical numbers of $f(x)=\left|x^3-3 x^2+2\right|$ correct to one decimal place.
$59-62$(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.$f(x)=x^5-x^3+2, \quad-1 \leqslant x \leqslant 1$
(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.$f(x)=x^4-3 x^3+3 x^2-x, \quad 0 \leqslant x \leqslant 2$
(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.$f(x)=x \sqrt{x-x^2}$
(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.$f(x)=x-2 \cos x, \quad-2 \leqslant x \leqslant 0$
Between $0^{\circ} \mathrm{C}$ and $30^{\circ} \mathrm{C}$, the volume $V$ (in cubic centimeters) of 1 kg of water at a temperature $T$ is given approximately by the formula$$V=999.87-0.06426 T+0.0085043 T^2-0.0000679 T^3$$Find the temperature at which water has its maximum density.
An object with weight $W$ is dragged along a horizontal plane by a 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(s) of friction and where $0 \leqslant \theta \leqslant \pi / 2$. Show that $F$ is minimized when $\tan \theta=\mu$.
A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function$$\begin{aligned}S(t)=- & 0.00003237 t^5+0.0009037 t^4-0.008956 t^3 \\& +0.03629 t^2-0.04458 t+0.4074\end{aligned}$$where $t$ is measured in years since August of 1993. Estimate the times when sugar was cheapest and most expensive during the period 1993-2003.
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.(TABLE CAN'T COPY)(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.
When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of 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 airstream, 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 airstream is related to the radius $r$ of the trachea by the equation$$v(r)=k\left(r_0-r\right) r^2 \quad \frac{1}{2} r_0 \leqslant r \leqslant r_0$$where $k$ is a constant and $r_0$ is the normal radius of the trachea. The restriction on $r$ is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than $\frac{1}{2} r_0$ is prevented (otherwise the person would suffocate).(a) Determine the value of $r$ in the interval $\left[\frac{1}{2} r_0, r_0\right]$ 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 $\left[0, r_0\right]$.
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 .
Prove that the function$$f(x)=x^{101}+x^{51}+x+1$$has neither a local maximum nor a local minimum.
If $f$ has a local minimum value at $c$, show that the function $g(x)=-f(x)$ has a local maximum value at $c$.
Prove Fermat's Theorem for the case in which $f$ has a local minimum at $c$.
A cubic function is a polynomial of degree 3 ; that is, it has the form $f(x)=a x^3+b x^2+c x+d$, where $a \neq 0$.(a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.(b) How many local extreme values can a cubic function have?