1. Critical Numbers – Relative Maximum and Minimum Points
Section 2.5
Critical Numbers – Relative Maximum and Minimum Points
Note: This is two day lecture, marked by and 2.5.2

2. If f ‘ (a) > 0
The graph of f(x) is INCREASING at x = a
x = a

3. If f ‘ (a) < 0
The graph of f(x) is DECREASING at x = a
x = a

4. If f ‘ (a) = 0, a maximum or minimum MAY exist.
This is the graph of g(x)
This is the graph of f(x)

5. If f ‘ (a) = 0, a maximum or minimum MAY exist.
This is the graph of f ‘ (x)
This is the graph of g ‘ (x)

6. A change from increasing to decreasing indicates a maximum

7. A change from decreasing to increasing indicates a minimum

8. FACTS ABOUT f ‘ (x) = 0
If f ‘ (x) > 0 on an interval (a, b), f is increasing on that interval
If f ‘ (x) < 0 on an interval (a, b), f is decreasing on that interval
If f ‘ (c) = 0 or f ‘ (c) does not exist, c is a critical number
If f ‘ (c) = 0, a relative max or min will exist IF f ‘ (x) changes from positive to negative OR 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.

9. This is the graph of f(x) on the interval [-1, 5].
Where are the relative extrema of f(x)?
x = -1, x = 1, x = 3, x = 5
For what value(s) of x is
f ‘ (x) < 0?
(1, 3)
For what value(s) of x is
f ‘ (x) > 0?
(-1, 1) and (3, 5)
D. Where are the zero(s) of f(x)?
x = 0
This is the graph of f(x)
on the interval [-1, 5].

10. This is the graph of f ‘ (x) on the interval [-1, 5]. (-1, 1), (3, 5)
Where are the relative extrema of f(x)?
x = -1, x = 0, x = 5
B. For what values of x is
f ‘ (x) < 0?
[-1, 0)
C. For what values of x is
f ‘ (x) > 0?
(0, 5]
For what values of x is
f “ (x) > 0?
This is the graph of f ‘ (x) on the
interval [-1, 5].
(-1, 1), (3, 5)

11. This is the graph of f(x) on [-10, 3].
Where are the relative extrema of f(x)?
x = -10, x = 3
On what interval(s) of x is
f ‘ (x) constant?
(-10, 0)
On what interval(s) is
f ‘ (x) > 0?
For what value(s) of x is
f ‘ (x) undefined?
x = -10, x = 0, x = 3
This is the graph of f(x) on [-10, 3].

12. This is the graph of f ‘ (x) on [-10, 3].
Where are the relative extrema of f(x)?
x = -10, x = -1, x = 3
On what interval(s) of x is
f ‘ (x) constant?
none
On what interval(s) is
f ‘ (x) > 0?
For what value(s) of x is
f ‘ (x) undefined?
none
This is the graph of f ‘ (x) on [-10, 3].

13. CALCULATOR REQUIRED
Based upon the graph of f ‘ (x) given
on the interval [0, 2pi], answer the following:
Where does f achieve a minimum value? Round your answer
to three decimal places.
3.665, 6.283
Where does f achieve a maximum value? Round your answer
to three decimal places.
0, 5.760

14. 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(x).
-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

15. 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(x).
-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

16. CALCULATOR REQUIRED
a) For what value(s) of x will there be a horizontal tangent? 1
b) For what value(s) of x will the graph be increasing?
c) For what value(s) of x will there be a relative minimum? 1
d) For what value(s) of x will there be a relative maximum?
none

17. This is the graph of f(x) on
For what value(s) of x is f ‘ (x) = 0?
On what interval(s) is f(x) increasing? .
Where are the relative maxima of f(x)?
-1 and 2
-1, 4
(-3, -1), (2, 4)
This is the graph of f(x) on
[-3, 4].

18. This is the graph of f ‘ (x)
For what value(s) of x if f ‘ (x) = 0?
For what value(s) of x does a relative maximum of f(x) exist?
For what value(s) of x is the
graph of f(x) increasing?
For what value(s) of x is the graph
of f(x) concave up?
-2, 1 and 3
-3, 1, 4
(-2, 1), (3, 4]
This is the graph of f ‘ (x)
[-3, 4]
(-3, -1) U (2, 4)

19. This is the graph of f(x) on
For what values of x if f ‘(x) undefined?
For what values of x is f(x) increasing?
For what values of x is f ‘ (x) < 0?
Find the maximum value of f(x). 6
(-5, 1)
(1, 3)
-5, 1, 3
This is the graph of f(x) on
[-5, 3]

20. This is the graph of f ‘ (x)
For what value(s) of x is f ‘ (x)
undefined?
For what values of x is f ‘ (x) > 0?
On what interval(s) is the graph of
f(x) decreasing?
On what interval(s) is the graph of f(x) concave up?
(0, 7)
(-7, 0)
(0, 7]
none
This is the graph of f ‘ (x)
on [-7, 7].

21. This is the graph of f(x) on
For what value(s) of x is
f ‘ (x) = 0?
For what value(s) of x does a
relative minimum exist?
On what intervals is f ‘ (x) > 0?
f “ (x) > 0?
(-1, 1), (1, 2)
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
-2, -0.5, 1.5
-1.5, -0.5, 0.5, 1.5
This is the graph of f(x) on
[-2, 2].

22. This is the graph of f ‘ (x) on
For what value(s) of x is
f ‘ (x) = 0?
For what value(s) of x is there a
local minimum?
f ‘ (x) > 0?
f “ (x) > 0?
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
(-2, -1), (0, 1)
-2, 0, 2
-2, -1, 0, 1, 2
This is the graph of f ‘ (x) on
[-2, 2]