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Section 2.5Critical Numbers – Relative Maximum and Minimum Points
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If f ‘ (a) > 0
The graph of f(x) is INCREASING at x = a
The graph of f ' x is POSITIVE at x = a
f(x)
f '(x)x = a
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If f ‘ (a) < 0
The graph of f(x) is DECREASING at x = a
f(x)
f '(x)
The graph of f ' x is aNE t GATIVE x = a
x = a
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This is the graph of g(x)This is the graph of f(x)
If f ‘ (a) = 0, a maximum or minimum MAY exist.
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If f ‘ (a) = 0, a maximum or minimum MAY exist.
This is the graph of g ‘ (x)This is the graph of f ‘ (x)
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A change from increasing to decreasing indicates a maximum
Graph of f(x)
Graph of f ' x
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A change from decreasing to increasing indicates a minimum
Graph of f ' x
Graph of f x
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•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.
FACTS ABOUT f ‘ (x) = 0
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A. Where are the relative extrema of f?
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].
x = -1, x = 1, x = 3, x = 5
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A. Where are the relative extrema of f?
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?
(-1, 1), (3, 5)This is the graph of f ‘ (x) on the interval [-1, 5].
x = -1, x = 0, x = 5
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A. Where are the relative extrema of f?
B. On what interval(s) is f ‘ constant?
(-10, 0)
C. On what interval(s) is f ‘ > 0?
D. For what value(s) of x is f ‘ 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].
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A. Where are the relative extrema of f?
B. On what interval(s) is f ‘ constant?
none
C. On what interval(s) is f ‘ > 0?
D. For what value(s) of x is f ‘ undefined?
none
x = -10, x = -1, x = 3
1, 3
This is the graph of f ‘ (x) on [-10, 3].
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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(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
CALCULATOR REQUIRED
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Estimate to one decimal place the critical numbers of f.
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
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Estimate to one decimal place the critical numbers of f.
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
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3
2
x 1Given f ' x
x 3
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
CALCULATOR REQUIRED
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For what value(s) of x is f ‘ (x) = 0?
On what interval(s) is f increasing?
. Where are the relative maxima of f?
-1 and 2
x = -1, x = 4
(-3, -1), (2, 4)
This is the graph of f(x) on [-3, 4].
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For what value(s) of x if f ‘ (x) = 0?
For what value(s) of x does a relative maximum of f exist?
On what interval(s) is f increasing?
On what interval(s) is f concave up?
-2, 1 and 3
-3, 1, 4
(-2, 1), (3, 4]
(-3, -1), (2, 4)This is the graph of f ‘ (x)
[-3, 4]
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This is the graph of f(x) on [-5, 3]
For what values of x if f undefined?
On what interval(s) is f increasing?
On what interval(s) is f ‘ < 0?
Find the maximum value of f.
6
(-5, 1)
(1, 3)
-5, 1, 3
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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 ‘ > 0?
On what interval(s) is f decreasing?
On what interval(s) is f concave up?
(0, 7)
(-7, 0)
(0, 7]
none
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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 interval(s) is f ‘ > 0?
On what interval(s) is f “ > 0?
(-1, 0), (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
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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?
On what interval(s) is f ‘ > 0?
On what interval(s) is f “ > 0?
(-2, -1.5), (-0.5, 0.5), (1.5, 2)
(-2, -1), (0, 1)
-2, 0, 2
-2, -1, 0, 1, 2