Practice Problems
Answers
1. Local maximums at , , , and . Global maximums at and .
Local minimums at , , , and .
Global minimum at .
3. so which is defined for all values of . when so is a critical number. There are no endpoints.
The only critical number is , and the
only critical point is which is the global (and local) minimum.
5. so which is defined for all values of . when so the values are critical numbers. There are no endpoints.
has local and global maximums at , and global and local minimums at .
7. so which is defined for all values of . when and so and are critical numbers. There are no endpoints. The only critical points are which is a local maximum and which is a local minimum. When the interval is the entire real number line, this function does not have a global maximum or global minimum.
9. so which is defined for all values of . when and so and are critical numbers. There are no endpoints. The only critical points are which is a local maximum and which is a local minimum. When the interval is the entire real number line, this function does not have a global maximum or global minimum.
11. so which is defined for all values of . is always positve (why?) so
is never equal to . There are no endpoints. The function
is always increasing and has no critical numbers, no critical points, no local or global maximums or minimums.
13. so which is defined for all values of . when so is a critical number. There are no endpoints. The only critical point is which is a local and global maximum. When the interval is the entire real number line, this function does not have a local or global minimum.
15. See Fig. 3.1P15
17. on so which is defined for all values of . when so is a critical number. The endpoints are and which are also critical numbers. The critical points are which is the local and global minimum, which is a local and global maximum, and which is a local maximum.
19. on so which is defined for all values of . when so is a critical number. The endpoints are and which are also critical numbers. The critical points are which is a local and global maximum, which is not a local or global maximum or minimum, and which is a local and global minimum.
21. on so which is defined for all values of . when and so these are critical numbers. The endpoints and are also critical numbers. The critical points are which is a local and global minimum on , the point which is a local and global maximum on , and the point which is a local and global minimum on .
23. so which is defined for all values of . when and in the interval so each of these values is a critical number. The endpoints and are also critical numbers. The critical points are which is a local minimum, which is a local and global maximum, and which is a local and global minimum. ( too, but is not in the interval ).
25. so which is defined for all values of . when but is not in the interval so is a not a critcal number. The endpoints and are critical numbers. The critical points are which is a local and global maximum, and which is a local and global minimum.
27.
A maximum is attained when .
29. for
cubic units is the largest volume.
Smallest volume is which occurs when and .
31. (a) . The endpoints and two values of for which .
(b) . The endpoints.
(c) At most . The endpoints and the interior points for which . At least . The
endpoints.
33. (a) local minimum at
(b) no extrema at
(c) local maximum at
(d) no extrema at
37. If f does not attain a maximum on or does not attain a mimimum on , then must have a discontinuity on .
39 (a) yes,
(b) no
(c) yes,
(d) no
(e) yes,
41. (a) yes,
(b) yes,
(c) yes,
(d) yes,
(e) yes,