Chapter 10

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{3}-3 x+2\) on \((-\infty, \infty)\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=x^{3}-6 x+1 $$

Let \(f(x)=\frac{e^{x}}{x^{2}+1} .\) Find and classify the critical points.

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{3}-3 x+2\) on \([-5,5]\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=-x^{3}+3 \pi^{2} x $$

A gardener has a fixed length of fence to fence off her rectangular chili pepper garden. Show that if she wants to maximize the area of her garden, then her garden should be square.

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{3}-3 x+2\) on \([0,3]\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=x^{3}+\frac{9}{2} x^{2}-12 x+\frac{3}{2} $$

A gardener needs 90 square feet of land for her tomato plants. She will fence in a rectangular plot. The cost of the fencing increases with the length of the perimeter. Show that the cost of the fencing is minimum if she uses a square plot.

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{3}-3 x+2\) on \((0,3)\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=x^{5}-5 x $$

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. Our ultimate goal is to find the value of \(x\) that will maximize the volume of the box. (a) Express the volume \(V\) of the box as a function of \(x\) and determine the appropriate domain. (b) Use the sign \(V^{\prime}\) to make a very rough sketch of the graph of \(V\) on \((-\infty, \infty)\). Identify the portion of the graph that is appropriate for the context of the problem. (c) Find the value of \(x\) that will maximize the volume of the box.

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=-2 x^{3}+3 x^{2}+12 x+5\) on \((-\infty, \infty)\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=2 x^{4}+64 x $$

You want to cut a rectangular wooden beam from a cylindrical log 14 inches in diameter. The strength of the beam is proportional to the quantity \(h^{2} w\), where \(h\) and \(w\) are the height and width of the cross section of the beam; the larger the quantity \(h^{2} w\), the stronger the beam. Find the height and width of the strongest beam that can be cut from the log. (Hint: You will need to find a way of relating \(h\) and \(w\). Sketch the circular cross section and sketch in a line denoting the diameter of the log. By placing the diameter line appropriately, you should be able to produce a right triangle made of \(w, h\), and the diameter. This will enable you to relate the width and height by using the Pythagorean Theorem.)

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=-2 x^{3}+3 x^{2}+12 x+5\) on \([-3,4]\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=x^{6}+x^{4} $$

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{5}-20 x+5\) on \((-\infty, \infty)\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=x^{4}+4 x^{3}+2 $$

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{5}-20 x+5\) on \([-2,0]\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point. $$ f(x)=4 x^{-1}+2 x^{2} $$

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=x^{5}-20 x+5\) on \([0,2]\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point.$$ f(x)=e^{x}-x $$

(a) Use your knowledge of shifting, flipping, and stretching to graph the function \(f(x)=-2|x-2|+4\) (b) At what value of \(x\) does \(f(x)\) attain its maximum value? At this point, what is \(f^{\prime}(x) ?\) (c) Does \(f(x)\) have a minimum value? (d) Where on the interval \(3 \leq x \leq 8\) does \(f\) take on its maximum value? Its minimum value?

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=3 x^{4}-8 x^{3}+3\) on \((-\infty, \infty)\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point.$$ f(x)=x e^{x}-e^{x} $$

For each of the following functions, determine where the function is increasing and where it is decreasing. Find the \(x\) -coordinates of all local maxima and minima. (Give exact answers, not numerical approximations.) (a) \(f(x)=2 x^{3}-24 x+4\) (b) \(f(x)=x^{3}-3 x^{2}-9 x+2\)

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=3 x^{4}-8 x^{3}+3\) on \([-1,1]\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point.$$ f(x)=\frac{x^{5}}{5}-x^{4}+\frac{4}{3} x^{3}+2 $$

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=3 x^{4}-8 x^{3}+3\) on \((0,3)\)

In Problems 1 through 12: (a) Find all critical points. (b) Find \(f^{\prime \prime} .\) Use the second derivative, wherever possible, to determine which critical points are local maxima and which are local minima. If the second derivative test fails or is inapplicable, explain why and use an alternative method for classifying the critical point.$$ f(x)=3 x^{4}-8 x^{3}+6 x+1 $$

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=\frac{x^{3}}{3}+2 x+\frac{3}{x}\) on its natural domain. Why is \(x=0\) not a critical point?

Suppose that \(f\) is a continuous function and that \(f(3)=2, f^{\prime}(3)=0\), and \(f^{\prime \prime}(3)=3\). At \(x=3\), does \(f\) have a local maximum, a local minimum, neither a local maximum nor a local minimum, or is it impossible to determine? Explain your answer.

A tin can for garbanza beans is designed to be a cylinder with volume of 300 cubic centimeters. Denote the radius by \(r\) and the height by \(h\). The top and bottom are thicker than the sides; for the purposes of our model, we'll assume that they are made with a double thickness of aluminum. (a) Give an expression for the volume of the can. (b) Give an expression for the amount of material used. (Remember that the top and bottom of the can are two layers thick.) (c) Make the expression from part (b) into a function of \(r\) alone. (d) What radius minimizes the material used? (e) What are the dimensions of the 300 cubic-centimeter can that require the least amount of material?

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=\frac{x^{3}}{3}+2 x+\frac{3}{x}\) on \([-3,0)\)

A can for mandarin oranges is a cylinder with volume of 250 cubic centimeters. Denote the radius by \(r\) and the height by \(h .\) The material used for the top and bottom is stronger than that used for the sides. There is wasted material in constructing the top and bottom because they need to be cut from squares of metal and the scrap metal is not used. The manufacturers must pay for the material for the whole square from which the circle is cut. Suppose that the material for the top and bottom is three times as expensive as the material for the sides. What are the dimensions of the can that minimize the cost of the materials?

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=\frac{x^{3}}{3}+2 x+\frac{3}{x}\) on \((0,3]\)

Without using the graphing capabilities of your graphing calculator, sketch the following graphs. Label the \(x\) -coordinates of all peaks and valleys. Label exactly, not using a numerical approximation. (If the \(x\) -coordinate is \(\sqrt{2}\), it should be labeled \(\sqrt{2}\), not \(1.41421 .)\) Below the sketch of \(f\), sketch \(f^{\prime}(x)\), labeling the \(x\) -intercepts of the graph of \(f^{\prime}\). (You can use your graphing calculator to check your answers.) (a) \(f(x)=x(x-9)(x-3)\) (Start by looking at the \(x\) -intercepts. Then look at the sign of \(f^{\prime}(x)\) in order to determine where the graph of \(f\) is increasing and where it is decreasing.) (b) \(f(x)=-2 x(x-9)(x-3)\) (Conserve your energy! Think!) (c) \(f(x)=-2 x(x-9)(x-3)+18\) (Conserve your energy! Think!)

For each function: (a) Find all critical points on the specified interval. (b) Classify each critical point: Is it a local maximum, a local minimum, an absolute maximum, or an absolute minimum? (c) If the function attains an absolute maximum and/or minimum on the specified interval, what is the maximum and/or minimum value? \(f(x)=\frac{1}{x^{2}+4}\) on \((-\infty, \infty)\)

(a) The function \(g\) with domain \((-\infty, \infty)\) is continuous everywhere. We are told that \(g^{\prime}(\sqrt{5})=0 .\) Some of the scenarios below would allow us to conclude that \(g\) has a local minimum at \(x=\sqrt{5}\). Identify all such scenarios. i. \(g(\sqrt{5})=0, g(2)=1, g(3)=1\) ii. \(g(\sqrt{5})<0\) and \(g^{\prime}(x)>0\) for \(x>\sqrt{5}\). iii. \(g^{\prime \prime}(\sqrt{5})>0\) iv. \(g^{\prime \prime}(\sqrt{5})<0\) v. \(g^{\prime}(x)>0\) for \(x<\sqrt{5}\) and \(g^{\prime}(x)<0\) for \(x>\sqrt{5}\) vi. \(g^{\prime}(x)<0\) for \(x<\sqrt{5}\) and \(g^{\prime}(x)>0\) for \(x>\sqrt{5}\) vii. \(g^{\prime}(\sqrt{5})>0\) and \(g^{\prime \prime}(\sqrt{5})=0\) (b) The function \(h\) with domain \([-8,-3]\) has the following characteristics. \(h\) is continuous at every point in its domain. \(h^{\prime}(x)<0\) for \(-80\) for \((-4,-3) .\) \(h^{\prime}(-4)\) is undefined. What can you conclude about the local and absolute extrema of \(h ?\) Please say as much as you can given the information above.

Let \(f(x)=\frac{e^{x}}{x}\). (a) Find all critical points of \(f\). (b) Identify all local extrema. (c) Does \(f\) have an absolute maximum value? If so, where is it attained? What is its value? (d) Does \(f\) have an absolute minimum value? If so, where is it attained? What is its value? (e) Answer parts (c) and (d) if \(x\) is restricted to \((0, \infty)\).

The graph of \(f^{\prime}\left(\right.\) not \(f\), but \(f^{\prime}\) ) is a parabola with \(x\) -intercepts of \(-\pi\) and \(2 \pi\) and a \(y\) -intercept of \(-2\). (a) Draw a graph of \(f^{\prime}\). (b) Write an equation for \(f^{\prime}\). This equation should have no unknown constants. (c) On the graph you drew in part (a), go back and label the \(x\) - and \(y\) -coordinates of the vertex. (d) Find \(f^{\prime \prime}(x)\). (e) This part of the question asks about \(f\), not \(f^{\prime}\). i. Where does \(f\) have a local maximum? Explain your reasoning clearly and briefly. ii. Where does \(f\) have a local minimum? Explain your reasoning clearly and briefly. iii. Does \(f\) have an absolute maximum or minimum value? Explain. iv. The function \(f\) has a single point of inflection. What is the \(x\) -coordinate of this point of inflection? Suppose you are told that the \(y\) -coordinate of the point of inflection is \(-1\). Find the equation of the tangent line to the graph of \(f\) at its point of inflection.

In its first printing, the printed material on a typical page of Frank McCourt's Angela's Ashes was \(4 \frac{1}{2}\) inches by \(7 \frac{1}{2}\) inches, with \(\frac{1}{2}\) -inch margins on the top and sides of the page and a 1 -inch margin on the bottom. Assuming pages must hold \(33.75\) square inches of printed matter and have the margins specified, was this book laid out in such a way as to minimize the amount of paper per page? If not, what page dimensions would minimize the page area?

Consider the function \(f(x)=x^{5}-2 x^{4}-7\) restricted to the domain \([-1,1]\). Your reasoning for the questions below must be fully explained and be independent of a graphing calculator. (a) Find the absolute maximum value of \(f(x)\) on the interval \([-1,1]\) or explain why this is not possible. (b) Find the absolute minimum value of \(f(x)\) on the interval \([-1,1]\) or explain why this is not possible. (c) Find the absolute minimum value of \(f(x)\) on the open interval \((-1,1)\) or explain why this is not possible.

Let \(f(x)=x^{2} e^{-x}\). (a) Find all critical points of \(f\). (b) Classify the critical points. (c) Does \(f\) take on an absolute maximum value? If so, where? What is it? (d) Does \(f\) take on an absolute minimum value? If so, where? What is it?

Q-Tips \(^{\otimes}\) are a brand of cotton swabs each 3 inches long. You can purchase a pack of 300 of them in a plastic rectangular container backed in cardboard. In other words, the plastic forms an open box and the "lid" is cardboard. The width of the box is 3 inches. What should the length and depth be if the goal is to minimize the amount of plastic used? In order to hold 300 Q-Tips the box must have a volume of \(33.75\) square inches. In reality, such a box is \(7.5\) inches long and \(1.5\) inches deep. Has the amount of plastic been minimized?

Find and classify all critical points. Determine whether or not \(f\) attains an absolute maximum and absolute minimum value. If it does, determine the absolute maximum and/or minimum value. f(x)=\left(x^{2}-4\right) e^{x}

\(f(x)=\frac{1}{3} x^{3}-2 x-\frac{1}{x}\) (a) Find \(f^{\prime}\). (b) Find \(f^{\prime \prime}\). (c) Find all critical points. Which of these critical points are also stationary points? (d) Analyze the critical points. Are they local extrema? Global extrema? Points of inflection? (e) Is \(x=0\) the \(x\) -value of a critical point? Why or why not? (f) What is the absolute maximum value of the function? The absolute minimum value?

Find and classify all critical points. Determine whether or not \(f\) attains an absolute maximum and absolute minimum value. If it does, determine the absolute maximum and/or minimum value. f(x)=\frac{10 x}{x^{2}+1}

Find and classify all critical points. Determine whether or not \(f\) attains an absolute maximum and absolute minimum value. If it does, determine the absolute maximum and/or minimum value. \(f(x)=\frac{10 x^{2}}{x^{2}+1}\)