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= 24 f'(x) > 0 Relative minimum nomial), you should try the second-derivative i to compute or if f"(a)= 0, use the first-deriva the second-derivative test is inconclusive. K function f(x)= x³ is always increasing and though f(0) = 0 and "(0) = 0. The function f'(x) = 4x³ and f"(x) = 12x². So f'(0) = 0 Fig. 14, you have a local minimum at x = 0.1 test, since f' changes sign at x = 0 and goes Example 6 here should serve as a summa derstanding 2.3 rves in Fig. 15 could possibly be the graph of 02 2. Which of the tion of the for

= 24 f'(x) > 0 Relative minimum nomial), you should try the second-derivative i to compute or if f"(a)= 0, use the first-deriva the second-derivative test is inconclusive. K function f(x)= x³ is always increasing and though f(0) = 0 and "(0) = 0. The function f'(x) = 4x³ and f"(x) = 12x². So f'(0) = 0 Fig. 14, you have a local minimum at x = 0.1 test, since f' changes sign at x = 0 and goes Example 6 here should serve as a summa derstanding 2.3 rves in Fig. 15 could possibly be the graph of 02 2. Which of the tion of the for

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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