Download Calculus-Increasing/Decreasing Functions

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Calculus-Increasing/Decreasing Functions
1.
Let f ( x ) = x − 1 + x + x + 2 .
a.
Find where f is increasing, where it is decreasing and where it is constant.
b.
Sketch the graph of the function in the space at right.
Note the indicted window size.
[-3, 3] x [-2, 10]
2.
Determine whether the following statements are always true or are at least sometimes false
and explain your answer. Include a sketch in every answer. You should assume that f is
differentiable on its domain.
a.
If f is increasing on an interval (a, b), then f N(x) $ 0 for all x in (a, b).
b.
If f is increasing on an interval (a, b), then f N(x) > 0 for all x in (a, b).
c.
If f is increasing on an interval (a, b), then f N(x) > 0 for at least one x in (a, b).
Revised August 2, 2000
Steve Boast
Calculus-Increasing/Decreasing Functions
d.
If f N(x) $ 0 for all x in an interval (a, b), then f is increasing on this interval.
e.
If f N(x) … 0 for all x on an interval (a, b), then f has no relative extrema on this
interval.
f.
If f N(x) > 0 for all x in an interval (a, b), and f N(x) < 0 in an interval (b, c), then f has
a relative maximum at b.
g.
If f has an absolute maximum on an open interval (a, b), then it must be a relative
maximum as well.
h.
If f has two relative maxima on an interval I, then it must have at least one relative
minimum on that interval.
i.
There are functions for which f (x) < 0 for all real numbers x but f N(x) > 0 for all real
numbers x.
Revised August 2, 2000
Steve Boast
Page 2
Calculus-Increasing/Decreasing Functions
3.
4.
The graph of the derivatives of two functions f and g are given below.
a.
What are the maximum number of solutions of f (x) = 0? Explain
b.
What are the maximum number of solutions of g (x) = 0? Explain
c.
If g (x) = 0 has two solutions, what can you say about where these two solutions lie?
Justify your answer.
The graph at right is the graph of the derivative of a
function f.
a.
Find where f is increasing and where it is decreasing.
b.
Find all relative maximum(s) and minimum(s).
c.
If f (-3) = -2, sketch the graph of f on the same axes.
Revised August 2, 2000
Steve Boast
Page 3
Calculus-Increasing/Decreasing Functions
5.
Suppose f is a function for which f N(x) > 0 for all real numbers x and let g(x) = f (f (x)). Must
g be increasing for all real numbers? Explain your answer.
6.
Suppose f is a function for which f N(x) < 0 for all real numbers x and let g(x) = f (f (x)). Must
g be decreasing for all real numbers? Explain your answer.
7.
Sketch the graph of the function f whose derivative satisfies the properties given in the table.
x
(-4, -1)
-1
(-1, 1)
1
(1, 3)
3
(3, 4)
f N(x)
positive
0
negative
0
positive
0
negative
a.
Does a vertical translation of f affect its degree?
Explain
b.
What is the minimum degree of f ?
Explain
c.
Does a vertical translation of f affect the number of relative extrema? Explain
d.
Does a vertical translation of f affect the type of relative extrema? Explain
Revised August 2, 2000
Steve Boast
Page 4