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Section 1

Maximum and Minimum Values

Explain the difference between an absolute minimum and alocal minimum.

Suppose $f$ is a continuous function defined on a closedinterval $[a, b] .$(a) What theorem guarantees the existence of an absolutemaximum value and an absolute minimum value for $f ?$(b) What steps would you take to find those maximum andminimum values?

$3-4$ . For each of the numbers $a, b, c, d, r,$ and $s$ , state whetherthe function whose graph is shown has an absolute maximum orminimum, a local maximum or minimum, or neither a maximumnor a minimum.

$5-6=$ Use the graph to state the absolute and local maximumand minimum values of the function.

$7-10=$ Sketch the graph of a function $f$ that is continuouson $[1,5]$ and has the given properties.Absolute minimum at $2,$ absolute maximum at 3local minimum at 4

$7-10=$ Sketch the graph of a function $f$ that is continuouson $[1,5]$ and has the given properties.Absolute minimum at $1,$ absolute maximum at $5,$local maximum at $2,$ local minimum at 4

$7-10=$ Sketch the graph of a function $f$ that is continuouson $[1,5]$ and has the given properties.Absolute maximum at $5,$ absolute minimum at $2,$local maximum at $3,$ local minima at 2 and 4

$7-10=$ Sketch the graph of a function $f$ that is continuouson $[1,5]$ and has the given properties.$f$ has no local maximum or minimum, but 2 and 4 arecritical numbers

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

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

(a) Sketch the graph of a function on $[-1,2]$ that has anabsolute 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 anabsolute minimum.

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

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$$$f(x)=\frac{1}{2}(3 x-1), \quad x \leqslant 3$$

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$$$f(x)=2-\frac{1}{3} x, \quad x \geqslant-2$$

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$$$f(x)=\sin x, \quad 0 \leqslant x<\pi / 2$$

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$$$f(t)=\cos t, \quad-3 \pi / 2 \leqslant t \leqslant 3 \pi / 2$$

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$$$f(x)=\ln x, \quad 0<x \leqslant 2$$

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$

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

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$$$f(x)=1-\sqrt{x}$$

$15-22=$ Sketch the graph of $f$ by hand and use your sketch tofind the absolute and local maximum and minimum values of $f .$(Use the graphs and transformations of Sections $1.2 . )$$$f(x)=\left\{\begin{array}{ll}{4-x^{2}} & {\text { if }-2 \leqslant x<0} \\ {2 x-1} & {\text { if } 0 \leqslant x \leqslant 2}\end{array}\right.$$

$23-36=$ Find the critical numbers of the function.$$f(x)=4+\frac{1}{3} x-\frac{1}{2} x^{2}$$

$23-36=$ Find the critical numbers of the function.$$f(x)=x^{3}+6 x^{2}-15 x$$

$23-36=$ Find the critical numbers of the function.$$f(x)=2 x^{3}-3 x^{2}-36 x$$

$23-36=$ Find the critical numbers of the function.$$f(x)=2 x^{3}+x^{2}+2 x$$

$23-36=$ Find the critical numbers of the function.$$g(t)=t^{4}+t^{3}+t^{2}+1$$

$23-36=$ Find the critical numbers of the function.$$g(t)=|3 t-4|$$

$23-36=$ Find the critical numbers of the function.$$g(y)=\frac{y-1}{y^{2}-y+1}$$

$23-36=$ Find the critical numbers of the function.$$h(p)=\frac{p-1}{p^{2}+4}$$

$23-36=$ Find the critical numbers of the function.$$F(x)=x^{4 / 5}(x-4)^{2}$$

$23-36=$ Find the critical numbers of the function.$$g(x)=x^{1 / 3}-x^{-2 / 3}$$

$23-36=$ Find the critical numbers of the function.$$f(\theta)=2 \cos \theta+\sin ^{2} \theta$$

$23-36=$ Find the critical numbers of the function.$$g(\theta)=4 \theta-\tan \theta$$

$23-36=$ Find the critical numbers of the function.$$f(x)=x^{2} e^{-3 x}$$

$23-36=$ Find the critical numbers of the function.$$f(x)=x^{-2} \ln x$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=12+4 x-x^{2}, \quad[0,5]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=5+54 x-2 x^{3}, \quad[0,4]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=2 x^{3}-3 x^{2}-12 x+1, \quad[-2,3]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=x^{3}-6 x^{2}+5, \quad[-3,5]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=3 x^{4}-4 x^{3}-12 x^{2}+1, \quad[-2,3]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=\left(x^{2}-1\right)^{3}, \quad[-1,2]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(t)=t \sqrt{4-t^{2}}, \quad[-1,2]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=\frac{x}{x^{2}-x+1},[0,3]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(t)=2 \cos t+\sin 2 t, \quad[0, \pi / 2]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(t)=t+\cot \left(\frac{1}{2} t\right), \quad[\pi / 4,7 \pi / 4]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=x e^{-x^{2} / 8}, \quad[-1,4]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=x-\ln x, \quad\left[\frac{1}{2}, 2\right]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=\ln \left(x^{2}+x+1\right), \quad[-1,1]$$

$37-50=$ Find the absolute maximum and absolute minimumvalues of $f$ on the given interval.$$f(x)=x-2 \tan ^{-1} x, \quad[0,4]$$

If $a$ and $b$ are positive numbers, find the maximum valueof $f(x)=x^{a}(1-x)^{b}, 0 \leqslant x \leqslant 1$

Use a graph to estimate the critical numbers of$f(x)=\left|x^{3}-3 x^{2}+2\right|$ correct to one decimal place.

$53-56$(a) Use a graph to estimate the absolute maximum andminimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimumvalues.$$f(x)=x^{5}-x^{3}+2, \quad-1 \leqslant x \leqslant 1$$

$53-56$(a) Use a graph to estimate the absolute maximum andminimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimumvalues.$$f(x)=e^{x}+e^{-2 x}, \quad 0 \leqslant x \leqslant 1$$

$53-56$(a) Use a graph to estimate the absolute maximum andminimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimumvalues.$$f(x)=x \sqrt{x-x^{2}}$$

$53-56$(a) Use a graph to estimate the absolute maximum andminimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimumvalues.$$f(x)=x-2 \cos x, \quad-2 \leqslant x \leqslant 0$$

Between $0^{\circ} \mathrm{C}$ and $30^{\circ} \mathrm{C}$ , the volume $V$ (in cubic centimeters of 1 $\mathrm{kg}$ of water at a temperature $T$ is givenapproximately 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 maximumdensity.

An object with weight $W$ is dragged along a horizontalplane 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 of friction and where 0$\leqslant \theta \leqslant \pi / 2 .$ Show that $F$ is minimizedwhen $\tan \theta=\mu$

A model for the US average price of a pound of white sugarfrom 1993 to 2003 is given by the function$$S(t)=-0.00003237 t^{5}+0.0009037 t^{4}-0.008956 t^{3}+0.03629 t^{2}-0.04458 t+0.4074$$where $t$ is measured in years since August of $1993 .$ Estimatethe times when sugar was cheapest and most expensive during the period $1993-2003 .$

The Hubble Space Telescope was deployed April $24,1990,$by the space shuttle Discovery. A model for the velocity ofthe shuttle during this mission, from liftoff at $t=0$ untilthe solid rocket boosters were jettisoned at $t=126 \mathrm{s},$ isgiven by$$v(t)=0.001302 t^{3}-0.09029 t^{2}+23.61 t-3.083$$(in feet per second). Using this model, estimate the absolutemaximum and minimum values of the acceleration of theshuttle between liftoff and the jettisoning of the boosters.

When a foreign object lodged in the trachea (windpipe)forces a person to cough, the diaphragm thrusts upwardcausing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrowerchannel for the expelled air to flow through. For a givenamount of air to escape in a fixed time, it must move fasterthrough the narrower channel than the wider one. Thegreater the velocity of the airstream, the greater the force onthe foreign object. $X$ -rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normalradius during a cough. According to a mathematical modelof coughing, the velocity $v$ of the airstream is related to theradius $r$ 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 $\left[\frac{1}{2} r_{0}, r_{0}\right]$ atwhich $v$ has an absolute maximum. How does thiscompare 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 extremevalue at $5 .$

Prove that the function $f(x)=x^{101}+x^{51}+x+1$ hasneither 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 localminimum at $c .$

A cubic function is a polynomial of degree $3 ;$ that is, it hasthe 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 nocritical number(s). Give examples and sketches to illus-trate the three possibilities.(b) How many local extreme values can a cubic functionhave?