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Section 2.5 Day 2Critical 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’
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.
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
A. Where are the relative extrema of f(x)?
B. For what value(s) of x is f ‘ (x) < 0?
(1, 3)
C. 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 = 0This is the graph of f(x) on the interval [-1, 5].
x = -1, x = 1, x = 3, x = 5
A. Where are the relative extrema of f(x)?
B. For what values of x is f ‘ (x) < 0?
[-1, 0)
C. For what values of x is f ‘ (x) > 0?
(0, 5]
D. For what values of x is f “ (x) > 0?
[-1, 1), (3, 5]This is the graph of f ‘ (x) on the interval [-1, 5].
x = -1, x = 0, x = 5
A. Where are the relative extrema of f(x)?
B. On what interval(s) of x is f ‘ (x) constant?
(-10, 0)
C. On what interval(s) is f ‘ (x) > 0?
D. For what value(s) of x is f ‘ (x) undefined?
x = -10, x = 0, x = 3
x = -10, x = 3
10, 0 , 0,3
This is the graph of f(x) on [-10, 3].
A. Where are the relative extrema of f(x)?
B. On what interval(s) of x is f ‘ (x) constant?
none
C. On what interval(s) is f ‘ (x) > 0?
D. For what value(s) of x is f ‘ (x) undefined?
none
x = -10, x = -1, x = 3
1, 3
This is the graph of f ‘ (x) on [-10, 3].
Based upon the graph of f ‘ (x) given f x cos x 1 sinx
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 answerto three decimal places.
0, 5.760
CALCULATOR REQUIRED
Estimate to one decimal place the critical numbers of f(x).
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 two questions below.
,
-1.4, -0.4, 0.4, 1.6
Estimate to one decimal place the critical numbers of f(x).
Estimate to one decimal place the value(s) of x at which there is a relative maximum.
1.1
Given the graph of f ‘ (x) on to the right, answer the three questions below.
,
-1.9, 1.1, 1.8
Estimate to one decimal place the value(s) of x at which there is a relative minimum.
-1.9, 1.8
3
2
x 1dyGiven
dx x 3
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? 1,
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
CALCULATOR REQUIRED
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].
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 graphof f(x) concave up?
-2, 1 and 3
-3, 1, 4
(-2, 1), (3, 4]
[-3, -1) U (2, 4]
This is the graph of f ‘ (x)[-3, 4]
This is the graph of f(x) on [-5, 3]
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 [-7, 7].
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 off(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 [-2, 2].
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?
For what value(s) of x is 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]
For what value(s) of x is f ‘ (x) = 0?
For what value(s) of x is there alocal minimum?
For what value(s) of x is f ‘ (x) > 0?
For what value(s) of x is 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
