MA119-A Applied Calculus for Business Fall Homework 8 Solutions Due 10/27/ :30AM

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1 MA9-A Applied Calculus for Business 006 Fall Homework 8 Solutions Due 0/7/006 0:0AM. # Find the interval(s) where the function f () = + + is increasing and the interval(s) where it is decreasing. First, we nd where f 0 () = 0. If f 0 () = + = 0, then we have =. Therefore, we have onl one critical point =. There are two intervals in this case, ; and. B picking a test point for each interval and calculating its sign of f 0, we have Intervals ; Test Point 0 f 0 of the test point Sign of f 0 + Increasing or decreasing Decreasing Increasing # Find the interval(s) where the function f () = + is increasing and the interval(s) where it is decreasing.

2 First, we nd where f 0 () = 0. We have f 0 () = 6= 0. But, when = (+), f 0 () is unde ned. Therefore, we have onl one critical point =. There are two intervals in this case, ; and. B picking a test point for each interval and calculating its sign of f 0, we have Intervals ; Test Point f 0 of the test point Sign of f 0 Increasing or decreasing Decreasing Decreasing # Find the interval(s) where the function f () = is increasing and the interval(s) where it is decreasing. First, we nd where f 0 () = 0. We have f 0 () = ()( ) ( )() =. If f 0 () = 0, ( ) ( ) then we have = 0, or, = 0 or. Also, when =, f 0 () is unde ned. Therefore, we have three critical points = 0; ;. There are four intervals in this case, ( ; 0), (0; ), (; ) and (; ). B picking a test point for each interval and calculating its sign of f 0, we have Intervals ( ; 0) (0; ) (; ) (; ) Test Point f 0 of the test point Sign of f Increasing or decreasing Increasing Decreasing Decreasing Increasing

3 #8 Find the relative maima and relative minima, if an, of the function g () = First, we nd where g 0 () = 0. If g 0 () = + = 0, then we have =. Therefore, we have onl one critical point =. There are two intervals in this case, ; and ;. B picking a test point for each interval and calculating its sign of f 0, we have Intervals ; Test Point 0 g 0 of the test point Sign of g 0 + Increasing or decreasing Decreasing Increasing Since g 0 changes its sign from to + when across = relative minimum., we know that = is a

4 #66 Find the relative maima and relative minima, if an, of the function g () =. First, we nd where g 0 () = 0. We have g 0 () = ()( ) ()() = ( ). If g 0 () = 0, ( ) then we have = 0. It provides no solution for us. But, when =, g 0 () is unde ned. Therefore, we have two critical points = ;. There are three intervals in this case, ( ; ), ( ; ) and (; ). B picking a test point for each interval and calculating its sign of g 0, we have Intervals ( ; ) ( ; ) (; ) Test Point 0 g of the test point 9 9 Sign of g 0 Increasing or decreasing Decreasing Decreasing Decreasing Since g 0 never changes its sign, there is no relative maima or minima.

5 #7 Growth of Managed Services Almost half of companies let other rms manage some of their Web operations a practice called Web hosting. Managed services monitoring a customer s technolog services is the fastest growing part of Web hosting. Managed services sales are epected to grow in accordance with the function f (t) = 0:69t + 0:758t + 0: (0 6) where f (t) is measured in billions of dollars and t is measured in ears, with t = 0 corresponding to 999. (a) Find the interval where f is increasing and the interval where f is decreasing. (b) What does our result tell ou about sales in managed services from 999 through 005? (a) First, we nd where f 0 () = 0. If f 0 () = 0:98 + 0:758 = 0, then we have = 0: Notice that 0:80805 < 0. Therefore, is no critical point since 0 6. There are two intervals in this case, ( ; 0:80805) and ( 0:80805; ). B picking a test point for each interval and calculating its sign of f 0, we have Intervals ( ; 0:80805) ( 0:80805; ) Test Point 0 f 0 of the test point 0:80 0:758 Sign of f 0 + Increasing or decreasing Decreasing Increasing Therefore, we know that f is increasing in the interval 0 6. (b) This result tells us that the sales in managed services is increasing from 999 through # Determine where the function g () = + +

6 6 is concave upward and where it is concave downward. First, we nd where g 00 () = 0. We have g 0 () = + and g 00 () =. Since g 00 never equals zero, there is onl one interval ( ; ) in this case. B picking a test point for each interval and calculating its sign of f 0, we have. # Determine where the function Intervals ( ; ) Test Point 0 g 00 of the test point Sign of g 00 Concavit Concave downward g () = + is concave upward and where it is concave downward. First, we nd where g 00 () = 0. We have g 0 () = ()(+). Notice that g 00 never equals zero. But, when = (+) ()() = (+) and g 00 () = (+), g 00 () is unde ned. Therefore, we have two intervals in this case, ( ; ) and ( ; ). B picking a test point for each interval and calculating its sign of g 0, we have Intervals ( ; ) ( ; ) Test Point 0 g 00 of the test point Sign of g 00 + Concavit Concave upward Concave downward 5 5

7 7. #0 Determine where the function g () = ( ) is concave upward and where it is concave downward. First, we nd where g 00 () = 0. We have g 0 () = ( ) () = ( ) and g 00 () = ( ) = ( ) 9 = p. Notice that 9 ( ) g00 never equals zero. But, when =, g 00 () is unde ned. Therefore, we have two intervals in this case, ( ; ) and (; ). B picking a test point for each interval and calculating its sign of g 0, we have Intervals ( ; ) (; ) Test Point g 00 of the test point 9 9 Sign of g 00 Concavit Concave downward Concave downward # Find the in ection point(s), if an, of the function g () = 6. First, we nd where g 00 () = 0. We have g 0 () = 6 and g 00 () = 6. If g 00 = 0, we have = 0. Therefore, we have two intervals in this case, ( ; 0) and (0; ). B picking a test point for each interval and calculating its sign of g 0, we have Intervals ( ; 0) (0; ) Test Point g 00 of the test point 6 6 Sign of g 00 + Concavit Concave downward Concave upward

8 8 Since g 00 changes its sign when across = 0, we know that = 0 is an in ection point #6 Find the in ection point(s), if an, of the function f () = + 6. First, we nd where f 00 () = 0. We have f 0 () = 6 and f 00 () =. If f 00 = 0, we have = 0 or. Therefore, we have three intervals in this case, ( ; 0), (0; ) and (; ). B picking a test point for each interval and calculating its sign of g 0, we have Intervals ( ; 0) (0; ) (; ) Test Point f 00 of the test point Sign of f Concavit Concave upward Concave upward Concave upward Since f 00 changes its sign when across = 0 and, we know that = 0 and = are both in ection points.

9 #5 Find the relative etrema, if an, of the function g () = Use the second derivative test, of applicable. First, we nd where g 0 () = 0. We have g 0 () = +. If g 0 = 0, we have =. To use the second derivative test, we need to know g 00 () =. Since g 00 = > 0, we know that = is a relative minimum #6 Find the relative etrema, if an, of the function f () = +.

10 0 Use the second derivative test, of applicable. First, we nd where f 0 () = 0. We have f 0 () = ()( +) ()() = +. If f 0 = 0, ( +) ( +) we have = 0, or, =. To use the second derivative test, we need to know ( ) ( + ) ( + ) ( ( + ) ()) g 00 () = ( + ) = ( + ) Since g 00 () = < 0, we know that = is a relative maimum. Since g 00 ( ) = > 0, we know that = is a relative minimum #86 Forecasting Pro ts As a result of increasing energ costs, the growth rate of the pro t of the -r old Venice Glassblowing Compan has begun to decline. Venice s management, after consulting with energ eperts, decides to implement certain energ-conservation measures aimed at cutting energ bills. The general manager reports that, according to his calculations, the growth rate of Venice s pro t should be on the increase again within r. If Venice s pro t (in hundreds of dollars) t r from now is given b the function P (t) = t 9t + 0t + 50 (0 t 8) determine whether the general manager s forcast will be accurate. [Hint: Fint the in ection point of the function P and stud the concavit of P.] The growth rate of Venice s pro t is the rate of change of P. Thus, it s P 0 (t) = t 8t + 0 where 0 t 8. We want to know if this growth rate is increasing or decreasing. So, we check (P 0 ) 0 = P 00 (t) = 6t 8. If P 00 = 0, then t =. So, after ears, P 00 () = 6 > 0. It tells us that P 0 is increasing when t =. Therefore, the general menager s forcase should be accurate.