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Each of the graphs of the functions in Exercises 17-24 has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. 17. f(x)= x³ + 6x² + 9x 18. f(x)=x²-x² 19. f(x) = x³-12x 20. f(x)= x³ +9x-2

Each of the graphs of the functions in Exercises 17-24 has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. 17. f(x)= x³ + 6x² + 9x 18. f(x)=x²-x² 19. f(x) = x³-12x 20. f(x)= x³ +9x-2

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Q17&Q19 needed to be solved correctly in 1 hour in the order to get positive feedback These are easy questions so solve these please do.all correctly
156 CHAPTER 2 Applications of the Derivative t J'(x) < 0 S'(0)=0 J"(0)=0 Figure 14 In Examples 1 and 2 we used the first derivative test to locate the extreme points; but in the remaining examples we used the second-derivative test. In general, which test should we use? If "(x) is straightforward to compute (for example, when f(x) is a poly- nomial), you should try the second-derivative test first. If the second derivative is tedious to compute or if f"(a) = 0, use the first-derivative test. Remember that when f"(a) = 0, a alte the second-derivative test is inconclusive. Keep the following examples in mind. The function f(x) = x3 is always increasing and has no local maximum or minimum, even though f(0) = 0 and f(0) = 0. The function f(x) = x4 is shown in Fig. 14. We have f'(x) = 4x³ and f"(x) = 12x². So f'(0) = 0 and f(0) = 0, but, as you can see from Fig. 14, you have a local minimum at x = 0. The reason follows from the first-derivative test, since f' changes sign at x = 0 and goes from negative to positive. f'(x) > 0 Example 6 here should serve as a summary of the techniques introduced so far. y = 24 0 Relative minimum Check Your Understanding 2.3 1. Which of the curves in Fig. 15 could possibly be the graph of a function of the form f(x) = ax² + bx + c, where a 0? V.K. (a) Figure 15 (2- (mumis on pol) (a) Figure 16 (b) EXERCISES 2.3 Each of the graphs of the functions in Exercises 1-8 has one relative. maximum and one relative minimum point. Find these points using the first-derivative test. Use a variation chart as in Example 1. 1. f(x)= x3-27x 2. f(x)=x² - 6x² + 1 3. f(x) = -x³ + 6x2² - 9x + 1 4. f(x) = -6x³3x² + 3x - 3 5. f(x)=x²-x²+1 6. f(x)=x²-x+ 2 7. f(x) = -x³-12x² - 2 8. f(x) = 2x³ + 3x² - 3 Solutions can be found following the section exercises. 2. Which of the curves in Fig. 16 could be the graph of a func- tion of the form f(x) = ax³ + bx² + cx + d, where a 0? ar to Each of the graphs of the functions in Exercises 9-16 has one rela- tive extreme point. Plot this point and check the concavity there. به (c) 21 na ty M (b) (d) Y (d) Using only this information, sketch the graph. [Recall that if f(x) = ax²+bx+c, then f(x) has a relative minimum point when a>0 and a relative maximum point when a < 0.1 9. f(x) = 2x² - 8 11. f(x)=x²+x-4 13. f(x) = 1 + 6x-x-² 15. f(x)=x²-8x - 10 10. f(x)=x² 12. f(x)=-3x² + 12x + 2 14. f(x)=x² + 16. f(x)=x² + 2x - 5 Each of the graphs of the functions in Exercises 17-24 has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. 17. f(x)= x³ + 6x² + 9x 19. f(x)=x³-12x 18. f(x)=x²-x² 20. f(x)= x³ +9x-2
Transcribed Image Text:156 CHAPTER 2 Applications of the Derivative t J'(x) < 0 S'(0)=0 J"(0)=0 Figure 14 In Examples 1 and 2 we used the first derivative test to locate the extreme points; but in the remaining examples we used the second-derivative test. In general, which test should we use? If "(x) is straightforward to compute (for example, when f(x) is a poly- nomial), you should try the second-derivative test first. If the second derivative is tedious to compute or if f"(a) = 0, use the first-derivative test. Remember that when f"(a) = 0, a alte the second-derivative test is inconclusive. Keep the following examples in mind. The function f(x) = x3 is always increasing and has no local maximum or minimum, even though f(0) = 0 and f(0) = 0. The function f(x) = x4 is shown in Fig. 14. We have f'(x) = 4x³ and f"(x) = 12x². So f'(0) = 0 and f(0) = 0, but, as you can see from Fig. 14, you have a local minimum at x = 0. The reason follows from the first-derivative test, since f' changes sign at x = 0 and goes from negative to positive. f'(x) > 0 Example 6 here should serve as a summary of the techniques introduced so far. y = 24 0 Relative minimum Check Your Understanding 2.3 1. Which of the curves in Fig. 15 could possibly be the graph of a function of the form f(x) = ax² + bx + c, where a 0? V.K. (a) Figure 15 (2- (mumis on pol) (a) Figure 16 (b) EXERCISES 2.3 Each of the graphs of the functions in Exercises 1-8 has one relative. maximum and one relative minimum point. Find these points using the first-derivative test. Use a variation chart as in Example 1. 1. f(x)= x3-27x 2. f(x)=x² - 6x² + 1 3. f(x) = -x³ + 6x2² - 9x + 1 4. f(x) = -6x³3x² + 3x - 3 5. f(x)=x²-x²+1 6. f(x)=x²-x+ 2 7. f(x) = -x³-12x² - 2 8. f(x) = 2x³ + 3x² - 3 Solutions can be found following the section exercises. 2. Which of the curves in Fig. 16 could be the graph of a func- tion of the form f(x) = ax³ + bx² + cx + d, where a 0? ar to Each of the graphs of the functions in Exercises 9-16 has one rela- tive extreme point. Plot this point and check the concavity there. به (c) 21 na ty M (b) (d) Y (d) Using only this information, sketch the graph. [Recall that if f(x) = ax²+bx+c, then f(x) has a relative minimum point when a>0 and a relative maximum point when a < 0.1 9. f(x) = 2x² - 8 11. f(x)=x²+x-4 13. f(x) = 1 + 6x-x-² 15. f(x)=x²-8x - 10 10. f(x)=x² 12. f(x)=-3x² + 12x + 2 14. f(x)=x² + 16. f(x)=x² + 2x - 5 Each of the graphs of the functions in Exercises 17-24 has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. 17. f(x)= x³ + 6x² + 9x 19. f(x)=x³-12x 18. f(x)=x²-x² 20. f(x)= x³ +9x-2
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