4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase

12 Use the given graph of to find the following. (a) The open intervals on which is increasing. (b) The open intervals on which is decreasing. (c) The open intervals on which is concave upward. (d) The open intervals on which is concave downward. (e) The coordinates of the points of inflection.

12 Use the given graph of to find the following. (a) The open intervals on which is increasing. (b) The open intervals on which is decreasing. (c) The open intervals on which is concave upward. (d) The open intervals on which is concave downward. (e) The coordinates of the points of inflection.

Suppose you are given a formula for a function . (a) How do you determine where is increasing or decreasing?(b) How do you determine where the graph of is concave upward or concave downward? (c) How do you locate inflection points?

(a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circum - stances is it inconclusive? What do you do if it fails?

56 The graph of the derivative of a function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum?

56 The graph of the derivative of a function is shown. (a) On what intervals is increasing or decreasing? (b) At what values of x does have a local maximum or minimum?

7. In each part state the -coordinates of the inflection points of . Give reasons for your answers. (a) The curve is the graph of . (b) The curve is the graph of . (c) The curve is the graph of .

The graph of the first derivative of a function is shown. (a) On what intervals is increasing? Explain. (b) At what values of does have a local maximum or minimum? Explain. (c) On what intervals is concave upward or concave down ward? Explain. (d) What are the -coordinates of the inflection points of ? Why?

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx 2x 3  3x 2 36x f

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx 4x 3  3x 2 6x  1 fx

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x4 2x2  3 fx

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x x 2  1 fx

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx sin x  cos x, 0  x  2 fx

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx cos2 x 2 sin x 0  x  2 fx

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx e2x  ex fx x

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x 2 ln x fx

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x 2 x ln x fx x

918 (a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.fx x 4 ex fx

1921 Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer fx 1  3x 2 2x 3 fx

1921 Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you preferfx x 2 x 1 f

1921 Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you preferfx sx s 4 x fx

. (a) Find the critical numbers of . (b) What does the Second Derivative Test tell you about the behavior of at these critical numbers? (c) What does the First Derivative Test tell you?

. Suppose is continuous on . (a) If and , what can you say about ? (b) If and , what can you say about ?

2429 Sketch the graph of a function that satisfies all of the given conditions.Vertical asymptote , if , if , if , if

2429 Sketch the graph of a function that satisfies all of the given conditions.f0 f2 f4 0 f x  0

2429 Sketch the graph of a function that satisfies all of the given conditions.f 1 f 1 0 f x  0  x   1 f x  0

2429 Sketch the graph of a function that satisfies all of the given conditions.f x  0  x   2 f x  0  x   2 f 

2429 Sketch the graph of a function that satisfies all of the given conditions.f x  0  x   2 f x  0  x   2 f 

2429 Sketch the graph of a function that satisfies all of the given conditions.fx  0 fx  0 x f 

Suppose , and and for all . (a) Sketch a possible graph for . (b) How many solutions does the equation have? Why? (c) Is it possible that ? Why?

3132 The graph of the derivative of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of x does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of f.

3132 The graph of the derivative of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of x does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of f.

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx x 3 12x  2 y

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx 36x  3x 2 2x 3 fx

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx 2  2x 2 x 4 fx

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.tx 200  8x 3  x 4 fx

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.hx x  1 5 5x 2 tx 20

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.hx 5x 3 3x 5 hx

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.Fx xs6 x hx

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.Gx 5x 23 2x 53 Fx

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.Cx x 4 

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.fx lnx Cx

33 44 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)(c) to sketch the graph. Check your work with a graphing device if you have one.f 2 cos  cos 0   2 2

Sx x sin x 0  x  4 f

4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx 1  1 x 1 x 2 f

4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx x 2 4 x 2  4 fx

4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx sx 2  1 x fx

4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx e x 1 e x fx s

4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx e ln

4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx ln1 ln x fx x

4552 (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)(d) to sketch the graph of .fx earctan x fx

Suppose the derivative of a function is . On what interval is increasing?

Use the methods of this section to sketch the curve , where is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?

5556 (a) Use a graph of to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of at which increases most rapidly. Then find the exact value.fx x  1 sx 2  1 fx

5556 (a) Use a graph of to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of at which increases most rapidly. Then find the exact value.f x x 2 ex fx

5758 (a) Use a graph of to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.fx cos x  0  x  2 1

5758 (a) Use a graph of to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.fx x 3 x 2 4 fx c

5960 Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .fx x 4  x 3  1 sx 2  x  1 f

5960 Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .fx x 2 tan1 x 1  x 3 fx

A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.

Let be the temperature at time where you live and suppose that at time you feel uncomfortably hot. How do you feel about the given data in each case?

Let be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, or ? Is the graph of K concave upward or concave downward? Why?

Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, , and is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve.

The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation. For simplicity, lets scale the function so as to remove the factor and lets analyze the special case where . So we study the function (a) Find the asymptote, maximum value, and inflection points of . (b) What role does play in the shape of the curve? ; (c) Illustrate by graphing four members of this family on the same screen.

. Find a cubic function that has a local maximum value of at and a local minimum value of 0 at .

For what values of the numbers and does the function have the maximum value ?

(a) If the function has the local minimum value at , what are the values of and ? (b) Which of the tangent lines to the curve in part (a) has the smallest slope?

For what values of and is an inflection point of the curve ? What additional inflection points does the curve have?

Show that the curve has three points of inflection and they all lie on one straight line.

Show that the curves and touch the curve at its inflection points.

Show that the inflection points of the curve lie on the curve .

7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.(a) If and are concave upward on , show that is concave upward on . (b) If is positive and concave upward on , show that the function is concave upward on .

7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.(a) If and are positive, increasing, concave upward functions on , show that the product function is concave upward on . (b) Show that part (a) remains true if and are both decreasing. (c) Suppose is increasing and is decreasing. Show, by giving three examples, that may be concave upward, concave downward, or linear. Why doesnt the argument in parts (a) and (b) work in this case?

7476 Assume that all of the functions are twice differentiable and the second derivatives are never 0.Suppose and are both concave upward on . Under what condition on will the composite function be concave upward?

Show that for . [Hint: Show that is increasing on .]

(a) Show that for . (b) Deduce that for . (c) Use mathematical induction to prove that for and any positive integer ,

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three -intercepts , and , show that the -coordinate of the inflection point is . b x 1s3 a 2 9s3 f x x 3  ax 2  bx y 2 x 2  4 4x 2 y x sin x x1  x2  x3 3 x x1, x2 x

For what values of does the polynomial have two inflection points? One inflection point? None? Illustrate by graphing for several values of . How does the graph change as decreases?

Prove that if is a point of inflection of the graph of and exists in an open interval that contains , then . [Hint: Apply the First Derivative Test and Fermats Theorem to the function .]

Show that if , then , but is not an inflection point of the graph of .

Show that the function has an inflection point at but does not exist

Suppose that is continuous and , but . Does have a local maximum or minimum at ? Does have a point of inflection at ?

Suppose is differentiable on an interval and for all numbers in except for a single number . Prove that is increasing on the entire interval .

. For what values of is the function increasing on ?

The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions whose values at 0 are all 0 and, for (a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that has neither a local maximum nor a local minimum at 0, has a local minimum, and has a local maximum.