Chapter 11

Determine whether or not the expression given is a polynomial. (a) \(\frac{1}{\sqrt{2}} x+\sqrt{33} x^{2}+\frac{19}{11}\) (b) \(2 x^{2}+3 x^{-1}+5 x^{3}\) (c) \(2 x+x^{1 / 2}+5 x^{5}\) (d) \(\frac{2}{x}+\frac{2 x}{3}+1\) (e) \(5^{-1 / 2} x+3^{-1} x^{2}+\frac{1}{\pi^{2}-2}+2\) (f) \(\left(x^{2}+1\right)^{-1}\)

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f(x)\) has zeros at \(x=-2, x=3\), and \(x=0\).

Suppose you are given a polynomial expression in both factored and nonfactored form. When might you prefer one form over the other?

Construct a polynomial \(P(x)\) with the specified characteristics. Answers to these problems are not unique. A second degree polynomial with zeros at \(x=1\) and \(x=-3\).

Suppose a distance function is given by \(d(t)=1 / t\) for \(0.5 \leq t \leq 20\). (a) What is the average velocity over the interval from \(t=1\) to \(t=5\) ? (b) Is there a time at which the instantaneous velocity is the same as the average velocity over the interval from \(t=1\) to \(t=5 ?\) If so, find that time. (c) On the same set of axes, illustrate your answers to parts (a) and (b).

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f(x)\) has zeros at \(x=-1\) and \(x=2\) only. \(f(0)=1\).

Suppose \(P(x)\) is a polynomial whose derivative is \(P^{\prime}(x)=x^{2}(x+2)^{3}\). (a) What degree is \(P(x)\) ? (b) What are the critical points of \(P(x)\) ? (c) Does \(P(x)\) have an absolute minimum value? If so, where is it attained? Is it possible to find out what this minimum value is, if it exists? If yes, explain how; if no, explain why not.

Construct a polynomial \(P(x)\) with the specified characteristics. Answers to these problems are not unique. A third degree polynomial with zeros at \(x=-1,0\), and 5 .

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f(x)\) has only one zero. It is at \(x=1 . \lim _{x \rightarrow \infty} f(x)=-\infty\).

Let \(p(x)\) be a polynomial of degree \(n .\) What is the maximum number of points of inflection possible for the graph of \(p(x)\) ?

Construct a polynomial \(P(x)\) with the specified characteristics. Answers to these problems are not unique. A fourth degree polynomial with no zeros.

Let \(f(x)=\frac{x^{2}+1}{x^{2}}=1+\frac{1}{x^{2}}\) (a) Graph \(f\). (b) Find the following. i. \(\lim _{x \rightarrow \infty} f(x) \quad\) ii. \(\lim _{x \rightarrow 0} f(x) \quad\) iii. \(\lim _{x \rightarrow \infty} f^{\prime}(x)\) iv. \(\lim _{x \rightarrow 0^{+}} f^{\prime}(x) \quad\) v. \(\lim _{x \rightarrow 0^{-}} f^{\prime}(x)\) (c) Graph \(f^{\prime}(x)\). (d) Find \(f^{\prime \prime}(x)\). (e) Are your answers to all parts of this problem consistent? (If not, find your errors.)

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f\) has a local maximum at \(x=0\) and a local minimum at \(x=2\).

A company is producing a single product. \(P(x)\), the profit function, gives profit as a function of \(x\), the number of hundreds of items produced. Suppose \(P(0)=-200\) and \(P^{\prime}(x)=x^{2}(x-1)^{3}\) Sketch the graph of \(P\). Argue, using the sign of \(P^{\prime}(x)\), that the graph of \(P\) intersects the positive \(x\) -axis exactly once, i.e., for \(x>0\), that there is one and only one breakeven point and that, if production levels are high enough, the profit will remain positive and increase with increasing \(x\). The following questions will help guide you. (a) First, draw a rough sketch of the graph of \(P^{\prime}(x)\). (You need not determine precisely the position of the local minimum of \(P^{\prime} ;\) in other words, you need not take the second derivative- just use what you know about the intercepts and sign of \(P^{\prime}(x) .\) ) (b) Draw a number line and on it record the sign of \(P^{\prime}\). Above it indicate where \(P\) is increasing and decreasing. (c) Now, using the information that \(P(0)=-200\) along with the information from part (b), make a rough sketch of \(P\). You need not determine the positive \(x\) -intercept, just convincingly assert that it exists.

Graph \(f(x)=\frac{x^{3}+x^{2}}{x^{2}-4}\). This function has four local extrema. One you can locate exactly. Where is it? Approximate the other three using your graphing calculator. (Notice that depending upon the viewing window you choose it may be very difficult to realize this function has four local extrema! When you use the calculator, use it carefully.)

Suppose \(P(x)\) with domain \((-\infty, \infty)\) is a polynomial of degree 4 whose leading coefficient is \(-3\). For each statement given below, determine whether the statement is necessarily true, or possibly true, possibly false, or definitely false. Think carefully. This is a problem concerning both logic and polynomials. (a) \(\lim _{x \rightarrow \infty} P(x)=\infty\) (b) \(\lim _{x \rightarrow-\infty} P(x)=-\infty\) (c) \(P(x)\) has four zeros. (d) \(P(x)\) has at least one turning point. (e) \(P(x)\) has exactly two turning points. (f) \(P(x)\) has four critical points. (g) \(P(x)\) has an absolute maximum value. (h) \(P(x)\) has an absolute minimum value.

Construct a polynomial \(P(x)\) with the specified characteristics. Answers to these problems are not unique. A fifth degree polynomial with a zero of multiplicity two at \(x=9\) and zeros at \(x=0\), 3, and \(-e\)

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f\) is always increasing.

Suppose \(P(x)\) is a polynomial of degree 7 whose leading coefficient is \(2 .\) For each statement given below, determine whether the statement is necessarily true, or \- possibly true, possibly false, or definitely false. If you decide the statement is not necessarily true, explain your reasoning! (a) \(P(x)\) has at least one zero and at most seven zeros. (b) \(P^{\prime}(x)\) has no zeros. (c) \(P(x)\) has at least one point of inflection. (d) \(P(x)\) has five points of inflection.

Construct a polynomial \(P(x)\) with the specified characteristics. Answers to these problems are not unique. A third degree polynomial whose only zero is at \(x=\pi+1\), and whose \(y\) -intercept is 1 .

Graph the following, clearly labeling all \(x\) - and \(y\) -intercepts, vertical asymptotes, and horizontal asymptotes. (a) \(y=\frac{x^{2}-4}{x^{2}-3 x-4}\) (b) \(y=\frac{3 x^{2}}{(x-1)^{2}}\) (c) \(y=\frac{(x-1)(x-2)}{x(x-1)(x-3)(x+1)}\)

Give an example of a cubic function \(f(x)\) with the characteristic(s) specified. Your answer should be a formula, but a picture will be helpful. There may be many possible answers. \(f\) is always decreasing. \(f(3)=0\) and \(f(0)=2\).

Graph \(f(x)=\frac{x^{4}}{4}+x^{3}-5 x^{2}\). Indicate the \(x\) -coordinates of all local extrema and all points of inflection. What is the absolute minimum value of \(f ?\) The absolute maximum value?

Let \(f(x)=(x-a)(x-b)^{2}\), where \(a>b>0\). By looking at the sign of \(f\) you can show that \(f\) has a local maximum at \(x=b\). This problem asks you to verify this using the second derivative test. (a) Using the Product Rule, show \(f^{\prime}(b)=0\). (b) Use the second derivative test to show that \(f\) has a local maximum at \(x=b\).

Construct a polynomial \(P(x)\) with the specified characteristics. Determine whether or not the answer to the problem is unique. Explain and/or illustrate your answer. A fourth degree polynomial with zeros of multiplicity two at \(x=2\) and \(x=-3\), and a \(y\) -intercept of \(-2\).

At one point in Leo Tolstoy's novella The Death of Ivan Ilyich, the title character states that the amount of blackness (the opposite of goodness in this context) in his life is in "inverse ratio to the square of the distance from death." Let \(B(t)\) represent the amount of blackness in his life, where \(t\) measures the amount of time since his birth, and let \(t=D\) represent the time of his death. (a) Write an equation for \(B(t)\). (Your answer should include the constant \(D .\) ) (b) Your equation for \(B(t)\) should have an arbitrary constant in it. Can you determine the sign of this constant? (c) Sketch a graph of \(B(t) .\) Is it increasing or decreasing? Concave up or concave down? Label any \(t\) - or \(B\) -intercepts and any asymptotes.

According to postal rules, the sum of the girth and the length of a parcel may not exceed 108 inches. What is the largest possible volume of a rectangular parcel with a square girth? ("Girth" means the distance around something. A person with a large girth needs a big belt.)

Construct a polynomial \(P(x)\) with the specified characteristics. Determine whether or not the answer to the problem is unique. Explain and/or illustrate your answer. A fifth degree polynomial with zeros of multiplicity two at \(x=0\) and \(x=\pi\), and a zero at \(x=-2 ; \lim _{x \rightarrow \infty} P(x)=\infty\).

What characteristics might the graph of a rational function (a polynomial divided by a polynomial) have that the graph of a polynomial will not have?

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Let \(x\) be the length of the sides of the corner squares. Find the value of \(x\) that will maximize the volume of the box.

Construct a polynomial \(P(x)\) with the specified characteristics. Determine whether or not the answer to the problem is unique. Explain and/or illustrate your answer. A third degree polynomial whose only zero is at \(x=-1\) and such that \(\lim _{x \rightarrow \infty} P(x)=\infty\)

Graph \(f(x)=\frac{4}{x}+x\). (a) Find \(f^{\prime}(x)\). Make a number line, marking all points at which \(f^{\prime}\) is zero or undefined. Use the number line to indicate the sign of \(f^{\prime}\); above this indicate where the graph of \(f\) is increasing and where it is decreasing. Note: \(x=0\) is not a critical point, since \(f\) is undefined at \(x=0\). However, it is possible for the sign of \(f^{\prime}\) to change on either side of a point at which \(f^{\prime}\) is undefined, so \(x=0\) must be labeled on your number line. (b) Find \(f^{\prime \prime}(x)\). Make a number line, marking all points at which \(f^{\prime \prime}\) is zero and undefined. Use the number line to indicate the sign of \(f^{\prime \prime}\); above this indicate where the graph of \(f\) is concave up and where it is concave down. (c) Graph \(f(x)\). Label both the \(x\) - and \(y\) -coordinates of the local maxima and local minima. (d) Does \(f(x)\) have an absolute maximum value? If so, what is it? Does \(f(x)\) have an absolute minimum value? If so, what is it?

Without using a graphing calculator, sketch the following graphs. Label all local maxima and minima. Beside the sketch of \(f\), draw a rough sketch of \(f^{\prime}(x)\). (a) \(f(x)=x(x-3)(x+5)\) (b) \(f(x)=-2 x(x-3)(x+5)\) (c) \(f(x)=x^{3}+3 x^{2}-9 x\) (d) \(f(x)=x^{3}+3 x^{2}-9 x+1\)

Construct a polynomial \(P(x)\) with the specified characteristics. Determine whether or not the answer to the problem is unique. Explain and/or illustrate your answer. The graph is a parabola with a vertex at \((\pi, 2)\) and a \(y\) -intercept of 0 .

Graph each of the following equations without using calculus. Label the following. (a) The \(x\) -intercepts; the \(y\) -intercepts (b) The vertical asymptotes (c) The horizontal asymptotes An analysis of where \(y\) is positive and where it is negative must be included. You need not find the coordinates of the local extrema. You need not look at \(y^{\prime}\). i. \(\quad y=\frac{x}{(x-1)(x+1)}\) ii. \(\quad y=\frac{x^{2}(x-2)}{(x-1)(x+1)}\) iii. \(y=\frac{x^{2}(x-2)}{(x-1)(x+1)}\) iv. \(y=\frac{x^{2}(x-2)}{(x-1)^{2}(x+1)}\) v. \(\quad y=\frac{x(x-2)}{(x-1)^{2}(x+1)}\) vi. \(y=\frac{(x-3)(x-2)}{(x-1)(x+1)}\) vii. \(y=\frac{2}{x^{2}+1}\) viii. \(y=\frac{-x^{2}}{x^{2}+1}\)

Find and classify all critical points. (a) \(f(x)=x^{3}-3 x+1\) (b) \(f(x)=x^{3}+3 x+1\)

(a) Suppose \(P(x)\) is a polynomial of degree \(5 .\) Which of the statements that follow must necessarily be true? If a statement is not necessarily true, provide a counterexample (an example for which the statement is false). i. \(P(x)\) has at least one zero. ii. \(P(x)\) has no more than four zeros. iii. The graph of \(P(x)\) has at least one turning point. iv. The graph of \(P(x)\) has at most four turning points. (b) Suppose \(P(x)\) is a polynomial of degree 5 with its natural domain \((-\infty, \infty)\). If \(P^{\prime}(\pi)=0\) and \(P^{\prime \prime}(\pi)=5\), then which one of the following statements is true? Explain your answer. i. \(P\) has a local minimum at \(x=\pi\) but this local minimum is not an absolute minimum. ii. \(P\) has a local minimum at \(x=\pi\) and this local minimum may be an absolute minimum. iii. \(P\) has a local maximum at \(x=\pi\) but this local maximum is not an absolute maximum. iv. \(P\) has a local maximum at \(x=\pi\) and this local maximum may be an absolute maximum.

Construct a polynomial \(P(x)\) with the specified characteristics. Determine whether or not the answer to the problem is unique. Explain and/or illustrate your answer. A fifth degree polynomial with a zero of multiplicity 3 at \(x=0\) and zeros at \(x=1\) and \(x=-2\), and passing through the point \((-1,2)\).

Graph the following functions using the information provided by the derivatives for guidance. Indicate where the function is increasing, where it is decreasing, and the coordinates of all local extrema. (a) \(f(x)=x+\frac{1}{x}\) (b) \(g(x)=x-\frac{1}{x}\)

Find and classify all critical points. (a) \(f(x)=-x^{3}-3 x^{2}+9 x+5\) (b) \(f(x)=x^{3}+3 x^{2}+9 x+8\)

(a) Suppose \(P(x)\) is a polynomial of degree \(6 .\) Which of the statements that follow must necessarily be true? If a statement is not necessarily true, provide a counterexample (an example for which the statement is false). i. \(P(x)\) has at least one zero. ii. \(P(x)\) has no more than five zeros. iii. The graph of \(P(x)\) has at least one turning point. iv. The graph of \(P(x)\) has at most five turning points. (b) Suppose \(P(x)\) is a polynomial of degree 6 with its natural domain \((-\infty, \infty)\). If \(P^{\prime}(2)=0\) and \(P^{\prime \prime}(2)=-1\), then which one of the following statements is true? Explain your answer. i. \(P\) has a local minimum at \(x=2\) but this local minimum is not an absolute minimum. ii. \(P\) has a local minimum at \(x=2\) and this local minimum may be an absolute minimum. iii. \(P\) has a local maximum at \(x=2\) but this local maximum is not an absolute maximum. iv. \(P\) has a local maximum at \(x=2\) and this local maximum may be an absolute maximum.

Construct a polynomial \(P(x)\) with the specified characteristics. Determine whether or not the answer to the problem is unique. Explain and/or illustrate your answer. A third degree polynomial with zeros at \(x=1\) and \(x=2\), a turning point at \(x=1\), and a \(y\) -intercept of \(\sqrt{e}\).

Suppose that \(f(x)\) is a rational function with zeros at \(x=0\) and \(x=4\), vertical asymptotes at \(x=-2\) and \(x=3\), and a horizontal asymptote at \(y=5\). For each of the following functions, indicate the location of any (a) zeros. (b) vertical asymptotes. (c) horizontal asymptotes. If there is not enough information to answer part of any question, say so. i. \(g(x)=f(x-3)\) ii. \(h(x)=f(x)-3\) iii. \(j(x)=2 f(3 x)\) iv. \(k(x)=f\left(x^{2}\right)\)

Find and classify all critical points. \(f(x)=x^{3}+x^{2}+x+1\)

Find the (real) zeros of the polynomial given. (a) \(f(x)=2 x^{3}+2 x^{2}-12 x\) (b) \(g(x)=2 x^{3}+2 x^{2}+12 x\)

Find and classify all critical points. $$ f(x)=-2 x^{3}+x^{2}+7 $$

The functions that follow in this exercise are not polynomials. We ask you about their range, domain, and graphs with the goal of having you appreciate how nicely polynomial functions behave. For each of the following functions: (a) Determine the domain. (b) Determine the range. (c) Sketch a graph of the function. Do this using your knowledge of flipping, stretching, shrinking, shifting, and of graphing \(\frac{1}{f(x)}\); check your graph with your graphing calculator. Your answers to parts (a) and (b) ought to agree with your answer to part (c). You can use your answers to parts (a) and (b) to select an appropriate viewing window in your calculator. i. \(f(x)=\frac{5}{x+20} \quad\) (The basic shape, before shifts and stretches, is \(y=1 / x .)\) ii. \(g(x)=-2 \sqrt{x-100} \quad\) (The basic shape, before shifts and stretches, is \(y=\sqrt{x}\). iii. \(h(x)=\frac{1}{\sqrt{x+40}}\) (Graph \(y=\sqrt{x}\), shift, and then look at the reciprocal.) iv. \(j(x)=\frac{2}{(x-20)(x+30)}\) \((\) Graph \(y=(x-20)(x+30)\), then look at the reciprocal.)

Find the (real) zeros of the polynomial given. (a) \(f(x)=-x^{3}-x^{2}-5 x\) (b) \(g(x)=0.5 x^{4}-0.5\)

Find and classify all critical points. $$ f(x)=x^{3}+2 x^{2}+3 x+4 $$

Find the (real) zeros of the polynomial given. (a) \(P(x)=x^{3}-x^{2}-4 x+4\) (b) \(Q(x)=x^{3}-x^{2}+4 x-4\)