1. I can sketch the graph of f given the graph of f’
Section 2.5 Day 2 Critical Numbers – Relative Maximum and Minimum Points
I can determine the key components of a function given the it’s derivative both graphically and numerically.
I can sketch the graph of f given the graph of f’

2. Vocabulary
Critical Number: a number c in the interior of the domain of a function is called this if either
f ‘ (c) = 0 or f ‘ (c) does not exist
Critical Point: the point (c, f(c)) of the graph f.
Local (Relative) Maximum: occurs at the highest point (a, f(a)) if f (your y-value) is the largest value.
Local (Relative) Minimum: occurs at the lowest point (a, f(a)) if f (your y-value) is the smallest value.

3. More Vocabulary
Local Extreme Values: Collectively, local maximum and minimum values
Local Extreme Points: local maximum and minimum points
Points of Inflection: Where function changes concavity

4. 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].

5. 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]

6. 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].

7. 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].

8. 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

9. 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

10. 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

11. 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

12. 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].

13. 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]

14. 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]

15. 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].

16. 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].

17. 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]