04 - Intervals of Increase and Decrease
[Pages:4]Kuta Software - Infinite Calculus
Intervals of Increase and Decrease
Name___________________________________ Date________________ Period____
For each problem, find the x-coordinates of all critical points, find all discontinuities, and find the open intervals where the function is increasing and decreasing.
1) y = -x3 + 2x2 + 2
y 8
6
4
2
-8 -6 -4 -2 -2 -4 -6 -8
2 4 6 8x
2) y = x3 - 11x2 + 39x - 47
3) y = -x4 + 3x2 - 3
x2 4) y =
4x + 4
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-1-
Worksheet by Kuta Software LLC
3x2 - 3 5) y =
x3
2
6) y = (2x - 8) 3
5
2
7) y = - 1 (x - 4) 3 - 2(x - 4) 3 - 1
5
Critical thinking question: 8) If functions f and g are increasing on an interval, show that f + g is increasing on the same interval.
9) Give an example where functions f and g are increasing on the interval (-,), but where f - g is decreasing.
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-2-
Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus
Name___________________________________
Intervals of Increase and Decrease
Date________________ Period____
For each problem, find the x-coordinates of all critical points, find all discontinuities, and find the open intervals where the function is increasing and decreasing.
1) y = -x3 + 2x2 + 2
y 8
6
4
2
-8 -6 -4 -2 -2 -4 -6 -8
2 4 6 8x
4 Critical points at: x = 0, No discontinuities exist.
3
( ) ( ) 4
Increasing: 0,
Decreasing: (-, 0),
4 ,
3
3
2) y = x3 - 11x2 + 39x - 47
13 Critical points at: x = 3, No discontinuities exist.
3
( ) ( ) Increasing: (-, 3),
13 ,
Decreasing:
13 3,
3
3
3) y = -x4 + 3x2 - 3
66 Critical points at: x = - , 0, No discontinuities exist.
22
( ) ( ) ( ) ( ) 6
6
6
6
Increasing: - , - , 0,
Decreasing: - , 0 , ,
2
2
2
2
x2 4) y =
4x + 4
Critical points at: x = -2, 0 Discontinuity at: x = -1
Increasing: (-, -2), (0, ) Decreasing: (-2, -1), (-1, 0)
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-1-
Worksheet by Kuta Software LLC
3x2 - 3 5) y =
x3 Critical points at: x = - 3, 3 Discontinuity at: x = 0
Increasing: (- 3, 0), (0, 3) Decreasing: (-, - 3), ( 3, )
2
6) y = (2x - 8) 3
Critical point at: x = 4 No discontinuities exist.
Increasing: (4, ) Decreasing: (-, 4)
5
2
7) y = - 1 (x - 4) 3 - 2(x - 4) 3 - 1
5
Critical points at: x = 0, 4 No discontinuities exist.
Increasing: (0, 4) Decreasing: (-, 0), (4, )
Critical thinking question:
8) If functions f and g are increasing on an interval, show that f + g is increasing on the same interval.
We know that if
x< 1
x , then 2
f (x1) <
f (x2) and
g(x1) <
g( x2).
Therefore,
f (x1) + g(x1) < f (x2) + g(x2).
9) Give an example where functions f and g are increasing on the interval (-,), but where f - g is decreasing.
Many answers. Ex: f = x and g = 2x
Create your own worksheets like this one with Infinite Calculus. Free trial available at
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Worksheet by Kuta Software LLC
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