Chapter 4

Explain the difference between a relative maximum and an absolute maximum. Sketch a graph that illustrates a function with a relative maximum that is not an absolute maximum, and sketch another graph illustrating an absolute maximum that is not a relative maximum. Explain how these graphs satisfy the given conditions.

Give a graph of the polynomial and label the coordinates of the intercepts, stationary points, and inflection points. Check your work with a graphing utility. $$ p(x)=4 x^{3}-9 x^{4} $$

From Exercise 57 , the polynomial \(f(x)=x^{3}+b x^{2}+1\) has one inflection point. Use a graphing utility to reach a conclusion about the effect of the constant \(b\) on the location of the inflection point. Use \(f^{\prime \prime}\) to explain what you have observed graphically.

Suppose that the intensity of a point light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. Two point light sources with strengths of \(S\) and \(8 S\) are separated by a distance of \(90 \mathrm{cm}\). Where on the line segment between the two sources is the total intensity a minimum?

Given points \(A(2,1)\) and \(B(5,4),\) find the point \(P\) in the interval \([2,5]\) on the \(x\) -axis that maximizes angle \(A P B .\)

Give a graph of the polynomial and label the coordinates of the intercepts, stationary points, and inflection points. Check your work with a graphing utility. $$ p(x)=x\left(x^{2}-1\right)^{3} $$

Consider the family of curves \(y=x e^{-b x}(b>0)\) (a) Use a graphing utility to generate some members of this family. (b) Discuss the effect of varying \(b\) on the shape of the graph, and discuss the locations of the relative extrema and inflection points.

The lower edge of a painting, \(10 \mathrm{ft}\) in height, is \(2 \mathrm{ft}\) above an observer's eye level. Assuming that the best view is obtained when the angle subtended at the observer's eye by the painting is maximum, how far from the wall should the observer stand?

In each part: (i) Make a conjecture about the behavior of the graph in the vicinity of its \(x\) -intercepts. (ii) Make a rough sketch of the graph based on your conjecture and the limits of the polynomial as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). (iii) Compare your sketch to the graph generated with a graphing utility. $$ \begin{array}{l}{\text { (a) } y=x(x-1)(x+1) \quad \text { (b) } y=x^{2}(x-1)^{2}(x+1)^{2}} \\ {\text { (c) } y=x^{2}(x-1)^{2}(x+1)^{3} \text { (d) } y=x(x-1)^{5}(x+1)^{4}}\end{array} $$

Consider the family of curves \(y=e^{-b x^{2}}(b>0)\) (a) Use a graphing utility to generate some members of this family. (b) Discuss the effect of varying \(b\) on the shape of the graph, and discuss the locations of the relative extrema and inflection points.

Let \(y=1 /\left(1+x^{2}\right) .\) Find the values of \(x\) for which \(y\) is increasing most rapidly or decreasing most rapidly.

(a) Determine whether the following limits exist, and if so, find them: $$ \lim _{x \rightarrow+\infty} e^{x} \cos x, \quad \lim _{x \rightarrow-\infty} e^{x} \cos x $$ (b) Sketch the graphs of the equations \(y=e^{x}, y=-e^{x}\), and \(y=e^{x} \cos x\) in the same coordinate system, and label any points of intersection. (c) Use a graphing utility to generate some members of the family \(y=e^{a x} \cos b x(a>0 \text { and } b>0),\) and discuss the effect of varying \(a\) and \(b\) on the shape of the curve.

Find the relative extrema in the interval \(0

Consider the family of curves \(y=x^{n} e^{-x^{2} / n},\) where \(n\) is a positive integer. (a) Use a graphing utility to generate some members of this family. (b) Discuss the effect of varying \(n\) on the shape of the graph, and discuss the locations of the relative extrema and inflection points.

Find the relative extrema in the interval \(0

If an unknown physical quantity \(x\) is measured \(n\) times, the measurements \(x_{1}, x_{2}, \ldots, x_{n}\) often vary because of uncontrollable factors such as temperature, atmospheric pressure, and so forth. Thus, a scientist is often faced with the problem of using \(n\) different observed measurements to obtain an estimate \(\bar{x}\) of an unknown quantity \(x .\) One method for making such an estimate is based on the least squares principle, which states that the estimate \(\bar{x}\) least squares principle, which states that the estimate \(\bar{x}\) should be chosen to minimize $$ s=\left(x_{1}-\bar{x}\right)^{2}+\left(x_{2}-\bar{x}\right)^{2}+\cdots+\left(x_{n}-\bar{x}\right)^{2} $$ which is the sum of the squares of the deviations between the estimate \(\bar{x}\) and the measured values. Show that the estimate resulting from the least squares principle is $$ \bar{x}=\frac{1}{n}\left(x_{1}+x_{2}+\cdots+x_{n}\right) $$ that is, \(\bar{x}\) is the arithmetic average of the observed values.

Find the relative extrema in the interval \(0

Prove: If \(f(x) \geq 0\) on an interval and if \(f(x)\) has a max- imum value on that interval at \(x_{0},\) then \(\sqrt{f(x)}\) also has a maximum value at \(x_{0}\). Similarly for minimum values. [Hint: Use the fact that \(\sqrt{x}\) is an increasing function on the interval \([0,+\infty) .]\)

Find the relative extrema in the interval \(0

A rectangular plot of land is to be fenced off so that the area enclosed will be \(400 \mathrm{ft}^{2}\). Let \(L\) be the length of fencing needed and \(x\) the length of one side of the rectangle. Show that \(L=2 x+800 / x\) for \(x>0,\) and sketch the graph of \(L\) versus \(x\) for \(x>0 .\)

Writing Discuss the importance of finding intervals of possible values imposed by physical restrictions on variables in an applied maximum or minimum problem.

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=x \ln x$$

A box with a square base and open top is to be made from sheet metal so that its volume is 500 in'. Let \(S\) be the area of the surface of the box and \(x\) the length of a side of the square base. Show that \(S=x^{2}+2000 / x\) for \(x>0,\) and sketch the graph of \(S\) versus \(x\) for \(x>0 .\)

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$ f(x)=x \ln x $$

Suppose that the number of individuals at time \(t\) in a certain wildlife population is given by $$N(t)=\frac{340}{1+9(0.77)^{t}}, \quad t \geq 0$$ where \(t\) is in years. Use a graphing utility to estimate the time at which the size of the population is increasing most rapidly.

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$ f(x)=x^{2} e^{-2 x} $$

Suppose that the spread of a flu virus on a college campus is modeled by the function $$y(t)=\frac{1000}{1+999 e^{-0.9 t}}$$ where \(y(t)\) is the number of infected students at time \(t\) (in days, starting with \(t=0\) ). Use a graphing utility to estimate the day on which the virus is spreading most rapidly.

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$ f(x)=10 \ln x-x $$

Suppose that \(x=x_{0}\) is a point at which a function \(f\) is continuous but not differentiable and that \(f^{\prime}(x)\) approaches different finite limits as \(x\) approaches \(x_{0}\) from either side. Invent your own term to describe the graph of \(f\) at such a point and discuss the appropriateness of your term.

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval, and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=x^{4}-24 x^{2}+12 x, \quad-5 \leq x \leq 5 $$

Assuming that \(A, k,\) and \(L\) are positive constants, verify that the graph of \(y=L /\left(1+A e^{-k t}\right)\) has an inflection point at \(\left(\frac{1}{k} \ln A, \frac{1}{2} L\right)\)

Suppose that the graph of a function \(f\) is obtained using a graphing utility. Discuss the information that calculus techniques can provide about \(f\) to add to what can already be inferred about \(f\) from the graph as shown on your utility's display.

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval, and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=\sin \frac{1}{2} x \cos x, \quad-\pi / 2 \leq x \leq \pi / 2 $$

An approaching storm causes the air temperature to fall. Make a statement that indicates there is an inflection point in the graph of temperature versus time. Explain how the existence of an inflection point follows from your statement.

Use a CAS to graph \(f^{\prime}\) and \(f^{\prime \prime},\) and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=\frac{10 x^{3}-3}{3 x^{2}-5 x+8} $$

Explain what the sign analyses of \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\) tell us about the graph of \(y=f(x) .\)

Use a CAS to graph \(f^{\prime}\) and \(f^{\prime \prime},\) and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=\frac{\tan ^{-1}\left(x^{2}-x\right)}{x^{2}+4} $$

Use a CAS to graph \(f^{\prime}\) and \(f^{\prime \prime},\) and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=\sqrt{x^{4}+\cos ^{2} x} $$

Use a CAS to graph \(f^{\prime}\) and \(f^{\prime \prime},\) and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=x^{2}\left(e^{2 x}-e^{x}\right) $$

In each part, find \(k\) so that \(f\) has a relative extremum at the point where \(x=3 .\) $$ \text { (a) } f(x)=x^{2}+\frac{k}{x} \quad \text { (b) } f(x)=\frac{x}{x^{2}+k} $$

(a) Use a CAS to graph the function $$ f(x)=\frac{x^{4}+1}{x^{2}+1} $$ and use the graph to estimate the x-coordinates of the relative extrema. (b) Find the exact \(x\) -coordinates by using the CAS to solve the equation \(f^{\prime}(x)=0\)

Functions similar to $$ f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} $$ arise in a wide variety of statistical problems. (a) Use the first derivative test to show that \(f\) has a relative maximum at \(x=0,\) and confirm this by using a graphing utility to graph \(f\) (b) Sketch the graph of $$ f(x)=\frac{1}{\sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2} $$ where \(\mu\) is a constant, and label the coordinates of the relative extrema.

Let \(h\) and \(g\) have relative maxima at \(x_{0} .\) Prove or disprove: (a) \(h+g\) has a relative maximum at \(x_{0}\) (b) \(h-g\) has a relative maximum at \(x_{0} .\)

Sketch some curves that show that the three parts of the first derivative test (Theorem 4.2.3) can be false without the assumption that \(f\) is continuous at \(x_{0} .\)

Writing Discuss the relative advantages or disadvantages of using the first derivative test versus using the second derivative test to classify candidates for relative extrema on the interior of the domain of a function. Include specific examples to illustrate your points.

If \(p(x)\) is a polynomial, discuss the usefulness of knowing zeros for \(p, p^{\prime},\) and \(p^{\prime \prime}\) when determining information about the graph of \(p .\)