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UAE_Math_Grade_12_Vol_1_SE_718383_ch3
In exercises 37–40, sketch a graph with the given properties.
4. Suppose that f(t) is the amount of money in your bank ac- 13:38
count at time t. Explain in terms of spending and saving ′ ′
what would cause f(t) to be decreasing and concave down; 37. f(0) = 0, f (x) > 0 for x < −1 and −1 < x < 1, f (x) < 0 for
′′
′′
increasing and concave up; decreasing and concave up. x > 1, f (x) > 0 for x < −1, 0 < x < 1 and x > 1, f (x) < 0
for −1 < x < 0
′
′
′′
38. f(0) = 2, f (x) > 0 for all x, f (0) = 1, f (x) > 0 for x < 0,
′′
In exercises 1–8, determine the intervals where the graph of the f (x) < 0 for x > 0
given function is concave up and concave down, and identify in- ′
flection points. 39. f(0) = 0,f(−1) =−1,f(1) = 1,f (x) > 0 for x < −1 and 0 <
′′
′
x < 1,f (x) < 0 for −1 < x < 0 and x > 1,f (x) < 0 for x < 0
and x > 0
1. f(x) = x − 3x + 4x − 1 2. f(x) = x − 6x + 2x + 3
2
3
4
2
′
′
3. f(x) = x + 1∕x 4. f(x) = x + 3(1 − x) 1∕3 40. f(1) = 0, f (x) < 0 for x < 1, f (x) > 0 for x > 1,
′′
f (x) < 0 for x < 1 and x > 1
2
−1
5. f(x) = sin x − cos x 6. f(x) = tan (x ) ............................................................
7. f(x) = x 4∕3 + 4x 1∕3 8. f(x) = xe −4x
3
2
............................................................ 41. Show that any cubic f(x) = ax + bx + cx + d has one inflec-
tion point. Find conditions on the coefficients a−e that guar-
4
3
2
In exercises 9–14, find all critical numbers and use the Second antee that the quartic f(x) = ax + bx + cx + dx + e has
Derivative Test to determine all local extrema. two inflection points.
42. If f and g are functions with two derivatives for all x,
2
3
4
4
9. f(x) = x + 4x − 1 10. f(x) = x + 4x + 1 f(0) = g(0) = f (0) = g (0) = 0, f (0) > 0 and g (0) < 0, state
′
′
′′
′′
11. f(x) = xe −x 12. f(x) = e −x 2 as completely as possible what can be said about whether
2
2
13. f(x) = x − 5x + 4 14. f(x) = x − 1 f(x) > g(x) or f(x) < g(x).
x
x
............................................................ 43. Give an example of a function showing that the following
statement is false. If the graph of y = f(x) is concave down
In exercises 15–26, determine all significant features by hand for all x, the equation f(x) = 0 has at least one solution.
and sketch a graph. 44. Determine whether the following statement is true or false.
′′
If f(0) = 1, f (x) exists for all x and the graph of y = f(x) is
15. f(x) = (x + 1) 2∕3 16. f(x) = x ln x concave down for all x, the equation f(x) = 0 has at least one
2
17. f(x) = x 2 18. f(x) = x solution.
2
x − 9 x + 2 In exercises 45 and 46, estimate the intervals of increase and
−x
19. f(x) = sin x + cos x 20. f(x) = e sin x decrease, the locations of local extrema, intervals of concavity
and locations of inflection points.
21. f(x) = x 3∕4 − 4x 1∕4 22. f(x) = x 2∕3 − 4x 1∕3
45. y
2
23. f(x) = x|x| 24. f(x) = x |x|
20
√
x
25. f(x) = x 1∕5 (x + 1) 26. f(x) = √
1 + x
............................................................ 10
In exercises 27–36, determine all significant features (approxi-
mately if necessary) and sketch a graph. x
-3 -2 2 3
4
3
27. f(x) = x − 26x + x
4
28. f(x) = 2x − 11x + 17x 2
3
46. y
3
29. f(x) = √ 2x − 1
2
10
30. f(x) = √ x + 1
3
5
31. f(x) = x − 16x + 42x − 39.6x + 14
4
3
2
x
32. f(x) = x + 32x − 0.02x − 0.8x - 2 2 4
3
2
4
√ 2x - 5
33. f(x) = x x − 4 34. f(x) = √
2
x + 4
2
( 1 )
35. f(x) = tan −1 x − 1 36. f(x) = e −2x cos x - 10 Copyright © McGraw-Hill Education
2
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276 | Lesson 4-5 | Concavity and the Second Derivative Test