Chap6 Section3

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 6.3 Extrema and Models 593 Version: Fall 2007 6.3 Exercises In Exercises 1 - 8 , perform each of the following tasks for the given polynomial. i. Without the aid of a calculator, use an algebraic technique to identify the zeros of the given polynomial. Factor if necessary. ii. On graph paper, set up a coordinate system. Label each axis, but scale only the x -axis. Use the zeros and the end-behavior to draw a “rough graph” of the given polynomial with- out the aid of a calculator. iii. Classify each local extrema as a rela- tive minimum or relative maximum . Note: It is not necessary to find the coordinates of the relative extrema. Indeed, this would be difficult with- out a calculator. All that is required is that you label each extrema as a relative maximum or minimum. 1. p ( x ) = ( x + 6)( x 1)( x 5) 2. p ( x ) = ( x + 2)( x 4)( x 7) 3. p ( x ) = x 3 6 x 2 4 x + 24 4. p ( x ) = x 3 + x 2 36 x 36 5. p ( x ) = 2 x 3 + 5 x 2 42 x 6. p ( x ) = 2 x 3 3 x 2 44 x 7. p ( x ) = 2 x 3 + 4 x 2 + 70 x 8. p ( x ) = 6 x 3 21 x 2 + 90 x In Exercises 9 - 16 , perform each of the following tasks for the given polynomial. i. Use a graphing calculator to draw the Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 graph of the polynomial. Adjust the viewing window so that the extrema or “turning points” of the polynomial are visible in the viewing window. Copy the resulting image onto your home- work paper. Label and scale each axis with xmin, xmax, ymin, and ymax. ii. Use the maximum and/or minimum util- ity in your calculator’s CALC menu to find the coordinates of the extrema. Label each extremum on your home- work copy with its coordinates and state whether the extremum is a rel- ative or absolute maximum or mini- mum. 9. p ( x ) = x 3 8 x 2 5 x + 84 10. p ( x ) = x 3 + 3 x 2 33 x 35 11. p ( x ) = x 3 + 21 x 20 12. p ( x ) = x 3 + 5 x 2 + 12 x 36 13. p ( x ) = x 4 50 x 2 + 49 14. p ( x ) = x 4 29 x 2 + 100 15. p ( x ) = x 4 2 x 3 39 x 2 + 72 x + 108 16. p ( x ) = x 4 3 x 3 31 x 2 + 63 x + 90 17. A square piece of cardboard mea- sures 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 box with no top. Find the dimensions of the squares cut from each corner of the original piece of cardboard so that
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594 Chapter 6 Polynomial Functions Version: Fall 2007 Cherie maximizes the volume of the re- sulting box. Perform each of the follow- ing steps in your analysis. a) Set up an equation that determines the volume of the box as a function of x , the length of the edge of each square cut from the four corners of the cardboard. Include any pictures used to determine this volume func- tion. b) State the empirical domain of the function created in part ( a ). Use your calculator to sketch the graph of the function over this empirical domain.
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