04 - Intervals of Increase and Decrease - Precalculus & AP Calculus

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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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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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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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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