9. p(x) = x 3 8x 2 5x p(x) = x 3 + 3x 2 33x p(x) = x x p(x) = x 3 + 5x x p(x) = x 4 50x
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- Gordon Reynolds
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1 Section 6.3 Etrema and Models Eercises In Eercises 1-8, perform each of the following tasks for the given polnomial. i. Without the aid of a calculator, use an algebraic technique to identif the zeros of the given polnomial. Factor if necessar. ii. On graph paper, set up a coordinate sstem. Label each ais, but scale onl the -ais. Use the zeros and the end-behavior to draw a rough graph of the given polnomial without the aid of a calculator. iii. Classif each local etrema as a relative minimum or relative maimum. Note: It is not necessar to find the coordinates of the relative etrema. Indeed, this would be difficult without a calculator. All that is required is that ou label each etrema as a relative maimum or minimum. 1. p() = ( + 6)( 1)( 5) 2. p() = ( + 2)( 4)( 7) 3. p() = p() = p() = p() = p() = p() = In Eercises 9-16, perform each of the following tasks for the given polnomial. i. Use a graphing calculator to draw the graph of the polnomial. Adjust the viewing window so that the etrema or turning points of the polnomial are visible in the viewing window. Cop the resulting image onto our homework paper. Label and scale each ais with min, ma, min, and ma. ii. Use the maimum and/or minimum utilit in our calculator s CALC menu to find the coordinates of the etrema. Label each etremum on our homework cop with its coordinates and state whether the etremum is a relative or absolute maimum or minimum. 9. p() = p() = p() = p() = p() = p() = p() = p() = A square piece of cardboard measures 12 inches per side. Cherie cuts four smaller squares from each corner of the cardboard square, tossing the material aside. She then bends up the sides of the remaining cardboard to form an open bo with no top. Find the dimensions of the squares cut from each corner of the original piece of cardboard so that 1 Coprighted material. See:
2 594 Chapter 6 Polnomial Functions Cherie maimizes the volume of the resulting bo. Perform each of the following steps in our analsis. a) Set up an equation that determines the volume of the bo as a function of, the length of the edge of each square cut from the four corners of the cardboard. Include an pictures used to determine this volume function. function created in part (a). Use our calculator to sketch the graph of the function over this empirical domain. Adjust the viewing window so that all etrema are visible in the viewing window. c) Cop the image in our viewing window onto our homework paper. Label and scale each ais with min, ma, min, and ma. Use the maimum utilit to find the coordinates of the absolute maimum on the function s empirical domain. d) What are the measures of the four squares cut from each corner of the original cardboard? What is the maimum volume of the bo? 18. A rectangular piece of cardboard measures 8 inches b 12 inches. Schuler cuts four smaller squares from each corner of the cardboard square, tossing the material aside. He then bends up the sides of the remaining cardboard to form an open bo with no top. Find the dimensions of the squares cut from each corner of the original piece of cardboard so that Schuler maimizes the volume of the resulting bo. Perform each of the following steps in our analsis. a) Set up an equation that determines the volume of the bo as a function of, the length of the edge of each square cut from the four corners of the cardboard. Include an pictures used to determine this volume function. function created in part (a). Use our calculator to sketch the graph of the function over this empirical domain. Adjust the viewing window so that all etrema are visible in the viewing window. c) Cop the image in our viewing window onto our homework paper. Label and scale each ais with min, ma, min, and ma. Use the maimum utilit to find the coordinates of the absolute maimum on the function s empirical domain. d) What are the measures of the four squares cut from each corner of the original cardboard? What is the maimum volume of the bo? 19. Restrict the graph of the parabola = 4 2 /4 to the first quadrant, then inscribe a rectangle inside the parabola, as shown in the figure that follows. (, ) a) Epress the area of the inscribed rectangle as a function of.
3 Section 6.3 Etrema and Models 595 function defined in part (a). Use our calculator to graph the area function over its empirical domain. Adjust the window parameters so that all etrema are visible in the viewing window. c) Cop the image in our viewing window to our homework paper. Label and scale each ais with min, ma, min, and ma. Use the maimum utilit to find the coordinates of the absolute maimum on the function s empirical domain. Label our graph with this result. utilit to find the coordinates of the absolute maimum on the function s empirical domain. Label our graph with this result. d) What are the length of the base and height of the triangle of maimum area? d) What are the dimensions of the rectangle of maimum area? 20. Restrict the graph of the parabola = 4 2 /4 to the first quadrant, then inscribe a triangle inside the parabola, as shown in the figure that follows. (, ) a) Epress the area of the inscribed triangle as a function of. function defined in part (a). Use our calculator to graph the area function over its empirical domain. Adjust the window parameters so that all etrema are visible in the viewing window. c) Cop the image in our viewing window to our homework paper. Label and scale each ais with min, ma, min, and ma. Use the maimum
4 596 Chapter 6 Polnomial Functions 6.3 Answers Local Maimum Local Maimum ( 6,0) (1,0) (5,0) ( 5,0) (0,0) (7,0) 3. Local Maimum (2,0) (6,0) ( 2,0) 9. Relative ma: ( , ) Relative min: ( , ) Answers ma differ slightl due to roundoff 100 ( , ) 5. Local Maimum p()= ( , ) 100 ( 6,0) (0,0) (7/2,0)
5 Section 6.3 Etrema and Models Relative min: ( , ) Relative ma: ( , ) Answers ma differ slightl due to roundoff 100 ( , ) ( , ) 15. Absolute min: ( , ) Relative ma: ( , ) Relative min: ( , ) Answers ma differ slightl due to roundoff 500 p()= ( , ) ( , ) 100 p()= ( , ) Absolute min: ( 5, 576) Relative ma: (0, 49) Absolute min: (5, 576) Answers ma differ slightl due to roundoff 600 p()= a) V = (12 2) 2 b) [0, 6] c) Absolute ma: (2, 128) V 200 (0,49) (2,128) V ()=(12 2) 2 ( 5, 576) 600 (5, 576) d) Cut square 2 inches on a side to produce a bo having value 128 in 3.
6 598 Chapter 6 Polnomial Functions 19. a) A = (4 2 /4) b) [0, 4] c) Absolute ma: ( , ) A 10 ( , ) V ()=(4 2 /4) d) = , =
9. p(x) = x 3 8x 2 5x p(x) = x 3 + 3x 2 33x p(x) = x x p(x) = x 3 + 5x x p(x) = x 4 50x
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