Section 2.5
Critical Numbers – Relative Maximum and Minimum Points
If f ‘ (a) > 0
The graph of f(x) is INCREASING at x = a
f(x)
The graph of f '  x  is POSITIVE at x = a
f '(x)
x=a
If f ‘ (a) < 0
The graph of f(x) is DECREASING at x = a
f(x)
The graph of f '  x  is NEGATIVE at x = a
f '(x)
x=a
If f ‘ (a) = 0, a maximum or minimum
This is the graph of f(x)
MAY exist.
This is the graph of g(x)
If f ‘ (a) = 0, a maximum or minimum
This is the graph of f ‘ (x)
MAY exist.
This is the graph of g ‘ (x)
A change from increasing to decreasing indicates a maximum
Graph of f(x)
Graph of f '  x 
A change from decreasing to increasing indicates a minimum
Graph of f  x 
Graph of f '  x 
FACTS ABOUT f ‘ (x) = 0
•If f ‘ (x) > 0 on an interval (a, b), f is increasing on (a, b).
•If f ‘ (x) < 0 on an interval (a, b), f is decreasing on (a, b).
•If f ‘ (c) = 0 or f ‘ (c) does not exist, c is a critical number
•If f ‘ (c) = 0, a relative maximum will exist IF f ‘ (x) changes from
positive to negative.
•If f ‘ (c) = 0, a relative minimum will exist IF f ‘ (x) changes from
negative to positive.
•A RELATIVE max/min is a high/low point around the area.
•An ABSOLUTE max/min is THE high/low point on an interval.
A. Where are the relative
extrema of f?
x = -1, x = 1, x = 3, x = 5
B. On what interval(s) is f ‘ < 0?
(1, 3)
C. On what interval(s) is f ‘ > 0?
(-1, 1) and (3, 5)
D. Where are the zero(s) of f?
x=0
This is the graph of f(x)
on the interval [-1, 5].
A. Where are the relative
extrema of f?
x = -1, x = 0, x = 5
B. On what interval(s) is f ‘ < 0?
[-1, 0)
C. On what interval(s) is f ‘ > 0?
(0, 5]
D. On what interval(s) is f “ > 0?
This is the graph of f ‘ (x) on the
interval [-1, 5].
(-1, 1), (3, 5)
A. Where are the relative
extrema of f?
x = -10, x = 3
B. On what interval(s) is f ‘
constant?
(-10, 0)
C. On what interval(s) is
f ‘ > 0?
 10, 0 , 0,3 
D. For what value(s) of x is
f ‘ undefined?
This is the graph of f(x) on [-10, 3].
x = -10, x = 0, x = 3
A. Where are the relative
extrema of f?
x = -10, x = -1, x = 3
B. On what interval(s) is
f ‘ constant?
none
C. On what interval(s) is
f ‘ > 0?
 1, 3
D. For what value(s) of x is
f ‘ undefined?
This is the graph of f ‘ (x) on [-10, 3].
none
CALCULATOR REQUIRED
Based upon the graph of f ‘ (x) given f  x   cos x 1  sin x 
on the interval [0, 2pi], answer the following:
Where does f achieve a minimum value? Round your answer(s)
to three decimal places.
x = 3.665, x = 6.283
Where does f achieve a maximum value? Round your answer(s)
to three decimal places.
x = 0, x = 5.760
Given the graph of f(x)
on  ,   to the right,
answer the two
questions below.
Estimate to one decimal place the critical numbers of f.
-1.4, -0.4, 0.4, 1.6
Estimate to one decimal place the value(s) of x at which
there is a relative maximum.
-1.4, 0.4
Given the graph of f ‘ (x)
on  ,   to the right,
answer the three
questions below.
Estimate to one decimal place the critical numbers of f.
-1.9, 1.1, 1.8
Estimate to one decimal place the value(s) of x at which
there is a relative maximum.
1.1
Estimate to one decimal place the value(s) of x at which
there is a relative minimum.
-1.9, 1.8
CALCULATOR REQUIRED
Given f '  x  
 x  1
3
 x  3
2
a) For what value(s) of x will f have a horizontal tangent?
1
b) On what interval(s) will f be increasing?
1,  
c) For what value(s) of x will f have a relative minimum?
1
d) For what value(s) of x will f have a relative maximum?
none
For what value(s) of x is f ‘ (x) = 0?
-1 and 2
On what interval(s) is f increasing?
.
(-3, -1), (2, 4)
Where are the relative maxima of
f?
This is the graph of f(x) on
[-3, 4].
x = -1, x = 4
For what value(s) of x if f ‘ (x) = 0?
-2, 1 and 3
For what value(s) of x does a
relative maximum of f exist?
-3, 1, 4
On what interval(s) is f increasing?
(-2, 1), (3, 4]
This is the graph of f ‘ (x)
[-3, 4]
On what interval(s) is f concave
up?
(-3, -1), (2, 4)
For what values of x if f undefined?
-5, 1, 3
On what interval(s) is f increasing?
(1, 3)
On what interval(s) is f ‘ < 0?
(-5, 1)
Find the maximum value of f.
This is the graph of f(x) on
[-5, 3]
6
For what value(s) of x is f ‘ (x)
undefined?
none
For what values of x is f ‘ > 0?
(0, 7]
On what interval(s) is f
decreasing?
(-7, 0)
On what interval(s) is f concave
up?
This is the graph of f ‘ (x)
on [-7, 7].
(0, 7)
For what value(s) of x is
f ‘ (x) = 0?
-1.5, -0.5, 0.5, 1.5
For what value(s) of x does a
relative minimum exist?
-2, -0.5, 1.5
On what interval(s) is f ‘ > 0?
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
This is the graph of f(x) on
[-2, 2].
On what interval(s) is f “ > 0?
(-1, 0), (1, 2)
For what value(s) of x is
f ‘ (x) = 0?
-2, -1, 0, 1, 2
For what value(s) of x is there a
local minimum?
-2, 0, 2
On what interval(s) is f ‘ > 0?
(-2, -1), (0, 1)
This is the graph of f ‘ (x) on
[-2, 2]
On what interval(s) is f “ > 0?
(-2, -1.5), (-0.5, 0.5), (1.5, 2)