Chapter 4

A real estate firm can borrow money at \(5 \%\) interest per year and can lend the money out. If the amount of money it can lend is inversely proportional to the square of the interest rate at which it lends, what interest rate would maximize the firm's profit per year? (Hint: Let \(x\) be the loan interest rate. Notice that the profit is the product of the amount borrowed by the firm and the difference between the interest rates at which it lends and borrows.)

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ g(x)=\sqrt{16-x^{2}} $$

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=x^{3}-3 x $$

A company has a daily fixed cost of $$\$ 5000$$. If the company produces \(x\) units daily, then the daily cost in dollars for labor and materials is \(3 x\). The daily cost of equipment maintenance is $$x^{2} / 2,500,000.$$ What daily production minimizes the total daily cost per unit of production? (Hint: The cost per unit is the total \(\operatorname{cost} C(x)\) divided by \(x .\) )

Find all vertical and horizontal asymptotes of the graph of \(f\). You may wish to use a graphics calculator to assist you. $$ f(x)=\frac{\ln (1-x)}{\ln (1+x)} $$

Suppose \(c\) and \(d\) are not both 0 , and let $$ f(x)=\frac{a x+b}{c x+d} $$ Show that \(f\) has no critical numbers unless \(a d-b c=0\), in which case \(f\) is a constant function.

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=x^{3}+3 x $$

A company sells 1000 units of a certain product annually, with no seasonal fluctuations in demand. It always reorders the same number \(x\) of units, stocks unsold units until no more remain, and then reorders again. If it costs \(b\) dollars to stock one unit for 1 year and there is a fixed cost of \(c\) dollars each time the company reorders, how many units should be reordered each time to minimize the total annual cost of reordering and stocking? (Hint: The company will have an average inventory of \(x / 2\) units and must reorder \(1000 / x\) times per year. Find the annual stocking and reordering costs and minimize their sum.)

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ g(x)=1 /(x+3) $$

Find all vertical and horizontal asymptotes of the graph of \(f\). You may wish to use a graphics calculator to assist you. $$ f(x)=\frac{\ln \left(1+e^{x}\right)}{\ln \left(2+e^{3 x}\right)} $$

In cach of the following, draw the graph of a continuous function \(f\) having the given propertics. a. \(f\) is increasing and its graph is concave upward on \((-\infty, 0)\), and \(f\) is decreasing and its graph is concave downward on \((0, \infty)\). b. \(f\) is decreasing and its graph is concave upward on \((-\infty, 1), f\) is increasing and its graph is concave upward on \((1,2)\), and \(f\) is decreasing and its graph is concave upward on \((2, \infty)\). c. \(f\) is decreasing and its graph is concave upward on \((-\infty, 1), f\) is increasing and its graph is concave upward on \((1,2)\), and \(f\) is increasing and its graph is concave downward on \((2, \infty)\). d. \(f\) is decreasing and its graph is concave downward on \((-\infty, 0), f\) is increasing and its graph is concave downward on \((0,1), f\) is increasing and its graph is concave upward on \((1,5)\), and \(f\) is decreasing and its graph is concave downward on \((5, \infty)\)

Assume that \(f\) is defined on \(I\) and that \(g=-f\). Prove that \(f\left(x_{0}\right)\) is the maximum value of \(f\) on \(I\) if and only if \(g\left(x_{0}\right)\) is the minimum value of \(g\) on \(I\).

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=2 x^{3}-3 x^{2}-12 x+5 $$

A farmer wishes to employ tomato pickers to harvest 62,500 tomatoes. Each picker can harvest 625 tomatoes per hour and is paid \(\$ 6\) per hour. In addition, the farmer must pay a supervisor \(\$ 10\) per hour and pay the union \(\$ 10\) for each picker employed. a. How many pickers should the farmer employ to minimize the cost of harvesting the tomatoes? b. What is the minimum cost to the farmer?

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ g(x)=(x-2) /(x-1) $$

Find all vertical and horizontal asymptotes of the graph of \(f\). You may wish to use a graphics calculator to assist you. $$ f(x)=x e^{1 / x} $$

Let \(f(x)=1 /\left(1+x^{2}\right)\) and \(g(x)=e^{-x^{2} / 2}\). Plot the graphs of \(f\) and \(g\) on \([-2,2]\), and determine which graph is more concave near the point \((0,1)\). Then justify your answer by comparing \(f^{\prime \prime}(0)\) and \(g^{\prime \prime}(0)\).

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=x^{4}+4 x $$

Find all vertical and horizontal asymptotes of the graph of \(f\). You may wish to use a graphics calculator to assist you. $$ f(x)=\frac{x^{4 / 3}+x^{1 / 3}-2}{x^{4 / 3}-16} $$

Recall that a function \(f\) is even if \(f(-x)=f(x)\) for all \(x\), and \(f\) is odd if \(f(-x)=-f(x)\) for all \(x\). a. If \(f\) is even and its graph is concave upward on \((0, \infty)\), what is the concavity on \((-\infty, 0)\) ? b. If \(f\) is odd and its graph is concave upward on \((0, \infty)\), what is the concavity on \((-\infty, 0)\) ?

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=x^{5}-5 x $$

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ k(x)=1 /\left(x^{2}-4\right) $$

It follows from the Substitution Rule that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(1 / x)\) and \(\lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow 0^{-}} f(1 / x)\) Use these formulas to evaluate the limit. $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}+1}{3 x^{2}-5} $$

Use the formulas for the function and its first and second derivatives as an aid in sketching the graph of the given function. Note all relevant properties listed in Table 4.1. $$ \begin{aligned} &f(x)=\frac{x}{\sqrt{1-x}}, f^{\prime}(x)=\frac{2-x}{2(1-x)^{3 / 2}}, \text { and } \\ &f^{\prime \prime}(x)=\frac{4-x}{4(1-x)^{5 / 2}} \end{aligned} $$

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=\left(x^{2}-1\right)^{2} $$

It follows from the Substitution Rule that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(1 / x)\) and \(\lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow 0^{-}} f(1 / x)\) Use these formulas to evaluate the limit. $$ \lim _{x \rightarrow-\infty} \frac{4 x^{3}-9 x^{2}}{-7 x^{3}+17} $$

Show by giving an example that the graph of the function \(f g\) need not be concave upward on an open interval \(I\) even if the graph of \(f\) is concave upward on \(I\) and the graph of \(g\) is concave upward on \(I .\)

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=x^{2}(x+3)^{2} $$

It follows from the Substitution Rule that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(1 / x)\) and \(\lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow 0^{-}} f(1 / x)\) Use these formulas to evaluate the limit. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{1+x^{2}}}{x} $$

Let \(f(x)=x^{1 / 3}\). Show that \((0,0)\) is an inflection point of the graph of \(f\), although neither \(f^{\prime}(0)\) nor \(f^{\prime \prime}(0)\) exists. (Thus it is not absolutely necessary for cither the first or the second derivative to exist in order to have an inflection point.)

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=(x-2)^{2}(x+1)^{2} $$

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ f(t)=2 \cos t-t $$

It follows from the Substitution Rule that \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(1 / x)\) and \(\lim _{x \rightarrow-\infty} f(x)=\lim _{x \rightarrow 0^{-}} f(1 / x)\) Use these formulas to evaluate the limit. $$ \lim _{x \rightarrow-\infty} \frac{\sqrt{1+x^{2}}}{x} $$

Sketch the graph of the function, using the Newton-Raphson method where necessary to find approximate zeros, critical points, and inflection points. $$ f(x)=2 x^{4}+x^{3}+x $$

Let \(f(x)=x^{5}-c x^{3}\), where \(c\) is a constant. Show that the graph of \(f\) has an inflection point at \((0,0)\).

Suppose \(R(x), C(x)\), and \(P(x)\) denote the revenue, cost, and profit resulting from the manufacture and sale of \(x\) units of an item. Recall that $$ P(x)=R(x)-C(x) \quad \text { for } x \geq 0 $$ Assume that it is possible to make a maximum profit by manufacturing \(x_{0}\) units of the item. Show that if \(R\) and \(C\) are differentiable and \(x_{0}>0\), then \(R^{\prime}\left(x_{0}\right)=C^{\prime}\left(x_{0}\right)\) (that is, the marginal revenue at \(x_{0}\) equals the marginal cost at \(x_{0}\) ).

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=\sqrt{x-x^{2}} $$

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ f(t)=\sin t+\cos t $$

Let $$ g(x)=\frac{1}{20} x^{5}+\frac{1}{4} x^{4}+\frac{1}{2} c x^{2} $$ Use your graphics calculator (or computer) to graph \(g\) "| for \(c=-5, c=-4, c=-1, c=0\), and \(c=1\). What can you tell about the possible number of inflection points for the graph of \(g\) ?

A mass connected to a spring moves along the \(x\) axis so that its \(x\) coordinate at time \(t\) is given by $$ x(t)=\sin 2 t+\sqrt{3} \cos 2 t $$ What is the maximum distance of the mass from the origin?

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=e^{x}-x $$

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ f(x)=x e^{x} $$

Let $$ f(x)=\frac{x|x|}{x^{2}+1} $$ Show that the graph of \(f\) has two horizontal asymptotes, and determine them.

Plot the graph of \(f\) and note all asymptotes. Then use the zoom features of the calculator to approximate the relative extreme values.. $$ f(x)=x^{9}-x^{5}+x^{4}-x^{3} $$

Let \(n\) be a positive integer and \(f(x)=x^{n} .\) Show that the graph of \(f\) has at most one inflection point. Determine those values of \(n\) for which the inflection point exists, and find the inflection point.

Use the First Derivative Test or the Second Derivative Test to determine the relative extreme values, if any, of the function. Then sketch the graph of the function. $$ f(x)=\ln (1+x)+\ln (1-x) $$

Find the intervals on which the given Ir function is increasing and those on which it is decreasing. $$ f(x)=e^{x}-3 x $$

Let $$ f(x)=\frac{\sqrt{2+4 x^{2}}}{x} $$ Show that the graph of \(f\) has two horizontal asymptotes, and determine them.

Plot the graph of \(f\) and note all asymptotes. Then use the zoom features of the calculator to approximate the relative extreme values.. $$ f(x)=\frac{1}{x}+\frac{1}{x-1}+\frac{1}{x-2} $$

It is known that a polynomial of degree \(k\) can have at most \(k\) real zeros. Use this fact to determine the maximum number of inflection points of the graph of a polynomial of degree \(n\), where \(n \geq 2\).