SECTION 4. MAXIMUM AND MINIMUM VALUES 285 The values of f at the endpoints are f 0 0 and f 2 2 6.28 Comparing these four numbers and using the Closed Interval Method, we see that the absolute minimum value is f 3 3 s3 and the absolute maimum value is f 53 53 s3. The values from part (a) serve as a check on our work. EXAMPLE 0 The Hubble Space Telescope was deploed on April 24, 990, b the space shuttle Discover. A model for the velocit of the shuttle during this mission, from liftoff at t 0 until the solid rocket boosters were jettisoned at t 26 s, is given b vt 0.00302t 3 0.09029t 2 23.6t 3.083 (in feet per second). Using this model, estimate the absolute maimum and minimum values of the acceleration of the shuttle between liftoff and the jettisoning of the boosters. SOLUTION We are asked for the etreme values not of the given velocit function, but rather of the acceleration function. So we first need to differentiate to find the acceleration: at vt d dt 0.00302t 3 0.09029t 2 23.6t 3.083 0.003906t 2 0.8058t 23.6 We now appl the Closed Interval Method to the continuous function a on the interval 0 t 26. Its derivative is at 0.00782t 0.8058 The onl critical number occurs when at 0: t 0.8058 23.2 0.00782 Evaluating at at the critical number and at the endpoints, we have a0 23.6 at 2.52 a26 62.87 So the maimum acceleration is about 62.87 fts 2 and the minimum acceleration is about 2.52 fts 2. 4. Eercises. Eplain the difference between an absolute minimum and a local minimum. 2. Suppose f is a continuous function defined on a closed interval a, b. (a) What theorem guarantees the eistence of an absolute maimum value and an absolute minimum value for f? (b) What steps would ou take to find those maimum and minimum values?
286 CHAPTER 4 APPLICATIONS OF DIFFERENTIATION 3 4 For each of the numbers a, b, c, d, e, r, s, and t, state whether the function whose graph is shown has an absolute maimum or minimum, a local maimum or minimum, or neither a maimum nor a minimum. 3. 4. 5 6 Use the graph to state the absolute and local maimum and minimum values of the function. 5. 6. 0 a b c d e r s t 0 a b c d e r s =ƒ 0 =ƒ 0 7 0 Sketch the graph of a function f that is continuous on [, 5] and has the given properties. 7. Absolute minimum at 2, absolute maimum at 3, local minimum at 4 8. Absolute minimum at, absolute maimum at 5, local maimum at 2, local minimum at 4 t 9. Absolute maimum at 5, absolute minimum at 2, local maimum at 3, local minima at 2 and 4 0. f has no local maimum or minimum, but 2 and 4 are critical numbers. (a) Sketch the graph of a function that has a local maimum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maimum at 2 and is continuous but not differentiable at 2. (c) Sketch the graph of a function that has a local maimum at 2 and is not continuous at 2. 2. (a) Sketch the graph of a function on [, 2] that has an absolute maimum but no local maimum. (b) Sketch the graph of a function on [, 2] that has a local maimum but no absolute maimum. 3. (a) Sketch the graph of a function on [, 2] that has an absolute maimum but no absolute minimum. (b) Sketch the graph of a function on [, 2] that is discontinuous but has both an absolute maimum and an absolute minimum. 4. (a) Sketch the graph of a function that has two local maima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maima, and seven critical numbers. 5 30 Sketch the graph of f b hand and use our sketch to find the absolute and local maimum and minimum values of f. (Use the graphs and transformations of Sections.2 and.3.) 5. f 8 3, 6. f 3 2, 5 7. f 2, 0 2 8. f 2, 0 2 9. f 2, 0 2 20. f 2, 0 2 2. f 2, 3 2 22. f 2, 23. f t t, 24. f t t, 25. f sin, 26. f tan, 27. 28. 29. 30. f s f e f 2 4 f 2 0 t 0 t 2 2 4 2 5 2 if 0 2 if 2 3 if 0 2 2 if 0
SECTION 4. MAXIMUM AND MINIMUM VALUES 287 3 46 Find the critical numbers of the function. 3. f 5 2 4 32. f 3 2 68. f cos 2 sin, 0 2 33. f 3 3 2 24 34. f 3 2 35. st 3t 4 4t 3 6t 2 36. 37. 38. 39. tt 5t 23 t 53 40. tt st t 4. 43. 47 62 Find the absolute maimum and absolute minimum values of f on the given interval. 47. f 3 2 2 5, 48. f 3 3, 50. f 3 6 2 9 2, 5. f 4 2 2 3, 52. f 2 3, 53. f, 2 54. f 2 4, 2 4 55. f t ts4 t 2, 56. f t s 3 t 8 t, 57. f sin cos, 58. f 2 cos, 59. 60. 6. 62. t 2 3 F 45 4 2 f 2 cos sin 2 f e, f ln, f 3 ln, f e e 2, 4, 4, 3 0, 8, 4 63. If a and b are positive numbers, find the maimum value of f a b, 0. ; 64. Use a graph to estimate the critical numbers of correct to one decimal place. f 3 3 2 2, 2, 2, 0, 42. 44. 45. f ln 46. f e 2 49. f 2 3 3 2 2, 2, 3 0, 3 f z z z 2 z t 3 23 G s 3 2 t 4 tan 2, 3, 4 67. f s 2 69. Between 0C and 30C, the volume V (in cubic centimeters) of kg of water at a temperature T is given approimatel b the formula V 999.87 0.06426T 0.0085043T 2 0.0000679T 3 Find the temperature at which water has its maimum densit. 70. An object with weight W is dragged along a horizontal plane b a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is F where is a positive constant called the coefficient of friction and where 0. Show that F is minimized when tan. 2 W sin cos 7. A model for the food-price inde (the price of a representative basket of foods) between 984 and 994 is given b the function It 0.00009045t 5 0.00438t 4 0.0656t 3 It 0.4598t 2 0.6270t 99.33 where t is measured in ears since midear 984, so 0 t 0, and It is measured in 987 dollars and scaled such that I3 00. Estimate the times when food was cheapest and most epensive during the period 984 994. ; 72. On Ma 7, 992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocit data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Event Time (s) Velocit (fts) Launch 0 0 Begin roll maneuver 0 85 End roll maneuver 5 39 Throttle to 89% 20 447 Throttle to 67% 32 742 Throttle to 04% 59 325 Maimum dnamic pressure 62 445 Solid rocket booster separation 25 45 ; 65 68 (a) Use a graph to estimate the absolute maimum and minimum values of the function to two decimal places. (b) Use calculus to find the eact maimum and minimum values. 65. f 3 8, 3 3 66. f e 3, 0 (a) Use a graphing calculator or computer to find the cubic polnomial that best models the velocit of the shuttle for the time interval t 0, 25. Then graph this polnomial. (b) Find a model for the acceleration of the shuttle and use it to estimate the maimum and minimum values of the acceleration during the first 25 seconds.
SECTION 4.2 THE MEAN VALUE THEOREM 295 4.2 Eercises 4 Verif that the function satisfies the three hpotheses of Rolle s Theorem on the given interval. Then find all numbers c that satisf the conclusion of Rolle s Theorem.. f 2 4, 2. f 3 3 2 2 5, 3. f sin 2, 4. f s 6, 5. Let f 23. Show that f f but there is no number c in, such that f c 0. Wh does this not contradict Rolle s Theorem? 6. Let f 2. Show that f 0 f 2 but there is no number c in such that f c 0. Wh does this not contradict Rolle s Theorem? 7. Use the graph of f to estimate the values of c that satisf the conclusion of the Mean Value Theorem for the interval 0, 8., 0, 4 6, 0 =ƒ 0 8. Use the graph of f given in Eercise 7 to estimate the values of c that satisf the conclusion of the Mean Value Theorem for the interval, 7. ; 9. (a) Graph the function f 4 in the viewing rectangle 0, 0 b 0, 0. (b) Graph the secant line that passes through the points, 5 and 8, 8.5 on the same screen with f. (c) Find the number c that satisfies the conclusion of the Mean Value Theorem for this function f and the interval, 8. Then graph the tangent line at the point c, f c and notice that it is parallel to the secant line. ; 0. (a) In the viewing rectangle 3, 3 b 5, 5, graph the function f 3 2 and its secant line through the points 2, 4 and 2, 4. Use the graph to estimate the -coordinates of the points where the tangent line is parallel to the secant line. (b) Find the eact values of the numbers c that satisf the conclusion of the Mean Value Theorem for the interval 2, 2 and compare with our answers to part (a). 4 Verif that the function satisfies the hpotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisf the conclusion of the Mean Value Theorem.. f 3 2 2 5, 2. f 3, 3. f e 2, 4. f, 2 5. Let. Show that there is no value of c such that f 3 f 0 f c3 0. Wh does this not contradict the Mean Value Theorem? f, 4, 6. Let f. Show that there is no value of c such that f 2 f 0 f c2 0. Wh does this not contradict the Mean Value Theorem? 7. Show that the equation 2 3 4 5 0 has eactl one real root. 8. Show that the equation 2 sin 0 has eactl one real root. 9. Show that the equation 3 5 c 0 has at most one root in the interval 2, 2. 20. Show that the equation 4 4 c 0 has at most two real roots. 2. (a) Show that a polnomial of degree 3 has at most three real roots. (b) Show that a polnomial of degree n has at most n real roots. 22. (a) Suppose that f is differentiable on and has two roots. Show that f has at least one root. (b) Suppose f is twice differentiable on and has three roots. Show that f has at least one real root. (c) Can ou generalize parts (a) and (b)? 23. If f 0 and f 2 for 4, how small can f 4 possibl be? 24. Suppose that 3 f 5 for all values of. Show that 8 f 8 f 2 30. 25. Does there eist a function f such that f 0, f 2 4, and f 2 for all? 26. Suppose that f and t are continuous on a, b and differentiable on a, b. Suppose also that f a ta and f t for a b. Prove that f b tb. [Hint: Appl the Mean Value Theorem to the function h f t.] 27. Show that s 2 if 0.
296 CHAPTER 4 APPLICATIONS OF DIFFERENTIATION 28. Suppose f is an odd function and is differentiable everwhere. Prove that for ever positive number b, there eists a number c in b, b such that f c f bb. 29. Use the Mean Value Theorem to prove the inequalit sin a sin b a b 30. If f c (c a constant) for all, use Corollar 7 to show that f c d for some constant d. 3. Let f and t Show that f t for all in their domains. Can we conclude from Corollar 7 that f t is constant? if if for all a and b 0 0 32. Use the method of Eample 6 to prove the identit 33. Prove the identit 2 sin cos 2 2 arcsin 2 arctan s 0 34. At 2:00 P.M. a car s speedometer reads 30 mih. At 2:0 P.M. it reads 50 mih. Show that at some time between 2:00 and 2:0 2 the acceleration is eactl 20 mih. 35. Two runners start a race at the same time and finish in a tie. Prove that at some time during the race the have the same speed. [Hint: Consider f t tt ht, where t and h are the position functions of the two runners.] 36. A number a is called a fied point of a function f if f a a. Prove that if f for all real numbers, then f has at most one fied point. 2 4.3 How Derivatives Affect the Shape of a Graph D B A C 0 FIGURE Man of the applications of calculus depend on our abilit to deduce facts about a function f from information concerning its derivatives. Because f represents the slope of the curve f at the point, f, it tells us the direction in which the curve proceeds at each point. So it is reasonable to epect that information about f will provide us with information about f. What Does f Sa about f? To see how the derivative of f can tell us where a function is increasing or decreasing, look at Figure. (Increasing functions and decreasing functions were defined in Section..) Between A and B and between C and D, the tangent lines have positive slope and so f 0. Between B and C, the tangent lines have negative slope and so f 0. Thus, it appears that f increases when f is positive and decreases when f is negative. To prove that this is alwas the case, we use the Mean Value Theorem. Let s abbreviate the name of this test to the I/D Test. Increasing/Decreasing Test (a) If f 0 on an interval, then f is increasing on that interval. (b) If f 0 on an interval, then f is decreasing on that interval. Resources / Module 3 / Increasing and Decreasing Functions / Increasing-Decreasing Detector Proof (a) Let and 2 be an two numbers in the interval with 2. According to the definition of an increasing function (page 2) we have to show that f f 2. Because we are given that f 0, we know that f is differentiable on, 2. So, b the Mean Value Theorem there is a number c between and such that 2 f 2 f f c 2 Now f c 0 b assumption and 2 0 because 2. Thus, the right side of