For problems 1 4 the graph of a function is given. Determine the open
intervals on which the function increases and decreases.
1.
2.
3.
4.
For problems 5 7 the graph of the derivative of a function is given.
From this graph determine the open intervals in which the function increases and decreases.
5.
6.
7.
For problems 8 10 The known information about the derivative
of a function is given. From this information answer each of the following
questions.
(a) Identify the critical points of the function.
(b) Determine the open intervals on which the function increases
and decreases.
(c) Classify the critical points as relative maximums, relative
minimums or neither.
8.
9.
10.
For problems 11 28 answer each of the following.
(a) Identify the critical points of the function.
(b) Determine the open intervals on which the function increases
and decreases.
(c) Classify the critical points as relative maximums, relative
minimums or neither.
11.
12.
13.
14.
15.
16.
17.
18. on
19. on
20. on
21. on
22.
23.
24.
25.
26.
27.
28.
29. Answer each of the following questions.
(a) What is the minimum degree of a
polynomial that has exactly one relative extrema?
(b) What is the minimum degree of a
polynomial that has exactly two relative extrema?
(c) What is the minimum degree of a
polynomial that has exactly three relative extrema?
(d) What is the minimum degree of a
polynomial that has exactly n
relative extrema?
30. For some function, ,
it is known that there is a relative minimum at . Answer each of the following questions about
this function.
(a) What is the simplest form that the
derivative of this function? Note : There really are many possible forms of the
derivative so to make the rest of this problem as simple as possible you will
want to use the simplest form of the derivative.
(b) Using your answer from (a) determine the most general form
that the function itself can take.
(c) Given that find a function that will have a relative
minimum at . Note : There
are many possible answers here so just give one of them.
31. For some function, ,
it is known that there is a relative maximum at . Answer each of the following questions about
this function.
(a) What is the simplest form that the
derivative of this function? Note : There really are many possible forms of the
derivative so to make the rest of this problem as simple as possible you will
want to use the simplest form of the derivative.
(b) Using your answer from (a) determine the most general form
that the function itself can take.
(c) Given that find a function that will have a relative
maximum at . Note : There
are many possible answers here so just give one of them.
32. For some function, ,
it is known that there is a critical point at that is neither a relative minimum or a
relative maximum. Answer each of the
following questions about this function.
(a) What is the simplest form that the
derivative of this function? Note : There really are many possible forms of the
derivative so to make the rest of this problem as simple as possible you will
want to use the simplest form of the derivative.
(b) Using your answer from (a) determine the most general form
that the function itself can take.
(c) Given that find a function that will have a critical
point at that is neither a relative minimum or a
relative maximum. Note : There are many possible answers here so just
give one of them.
33. For some function, ,
it is known that there is a relative maximum at and a relative minimum at . Answer each of the following questions about
this function.
(a) What is the simplest form that the
derivative of this function? Note : There really are many possible forms of the
derivative so to make the rest of this problem as simple as possible you will
want to use the simplest form of the derivative.
(b) Using your answer from (a) determine the most general form
that the function itself can take.
(c) Given that and find a function that will have a relative
maximum at and a relative minimum at . Note : There
are many possible answers here so just give one of them.
34. Given that and are increasing functions will always be an increasing function? If so, prove that will be an increasing function. If not, find increasing functions, and ,
so that will be a decreasing function and find a
different set of increasing functions so that will be an increasing function.
35. Given that is an increasing function. There are several possible conditions that we
can impose on so that will be an increasing function. Determine as many of these possible
conditions as you can.
36. For a function determine a set of conditions on ,
different from those given in #15 in
the practice problems, for which will be an increasing function.
37. For a function determine a single condition on for which will be an increasing function.
38. Given that and are positive functions. Determine a set of
conditions on them for which will be an increasing function. Note that there are several possible sets of
conditions here, but try to determine the “simplest” set of conditions.
39. Repeat #38 for .
40. Given that and are increasing functions prove that will also be an increasing function.