Infinite Calculus - HW 5.3 Concavity & Points of Inflection

Calculus

Name___________________________________ ID: 1

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HW 5.3 Concavity & Points of Inflection

Date________________ Period____

For each problem, find the x-coordinates of all points of inflection and find the open intervals where the function is concave up and concave down.

1) f (x) = 2x2 - 12x + 20

2) f (x) = -x3 + 2x2 + 1

3) f (x) = x3 - 3x2 + 3

4) f (x) = x4 - x3 - 3x2 + 4

5) f (x) = 3

x + 1

6) f (x) = x2

2x + 2

7) f (x) = 3x

x + 1

8) f (x) = x

x + 1

For each problem, find the x-coordinates of all critical points and find the open intervals where the function is increasing and decreasing.

9) f (x) = -2x2 - 8x - 9

10) f (x) = x3 - 4x2 + 5

11) f (x) = -x4 + 2x2

12) f (x) = 1

x - 1

Worksheet by Kuta Software LLC

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For each problem, find the open intervals where the function is increasing and decreasing.

13) f (x) = 2x2 + 16x + 27

14) f (x) = x3 - x2 + 4

15) f (x) = x4 + 2x3 - 2x2 + 2

16) f (x) = 3

x + 2

dy For each problem, use implicit differentiation to find in terms of x and y.

dx

17) 3 y2 + 2 = 5x3

18) 3x3 + 5 y2 = 2

19) 3 y3 + y2 = 4x

20) 3x2 = 4 y2 + y

21) x2 - 3 y = 5x3y2

22) 3x2 - y2 = xy3

23) -3x3y3 + 1 = x

24) 2 = 2x2 + 3x3y3

For each problem, find the indicated derivative with respect to x.

25) f (x) = -5x5 - 3x3 + x2 Find f ''

26) f (x) = x4 - 5x2 + 5x Find f '''

27) f (x) = 5x3 + 5x2 - 4x Find f (4)

28) f (x) = 5x5 - 4x4 - 4x2 Find f '''

Differentiate each function with respect to x. -2x - 3

29) y =

(-2x2 + 5)-3

30) y = (3x2 - 1)-3(-x3 - 2)

Worksheet by Kuta Software LLC

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Answers to HW 5.3 Concavity & Points of Inflection (ID: 1)

1) No inflection points exist.

Concave up: (-?, ?) Concave down: No intervals exist.

2 2) Inflection point at: x =

3

3) Inflection point at: x = 1

Concave up: (1, ?) Concave down: (-?, 1)

( ) ( ) 2

2

Concave up: -?, Concave down: , ?

3

3

1 4) Inflection points at: x = - , 1

2

( ) ( ) Concave up:

1 -?, -

, (1, ?)

Concave down:

1 - ,1

2

2

5) No inflection points exist.

6) No inflection points exist.

Concave up: (-1, ?) Concave down: (-?, -1)

Concave up: (-1, ?) Concave down: (-?, -1)

7) No inflection points exist.

8) No inflection points exist.

Concave up: (-?, -1) Concave down: (-1, ?)

Concave up: (-?, -1) Concave down: (-1, ?)

9) Critical point at: x = -2

Increasing: (-?, -2) Decreasing: (-2, ?)

8 10) Critical points at: x = 0,

3

( ) ( ) Increasing: (-?, 0),

8 , ?

8 Decreasing: 0,

3

3

11) Critical points at: x = -1, 0, 1

Increasing: (-?, -1), (0, 1) Decreasing: (-1, 0), (1, ?)

12) No critical points exist.

Increasing: No intervals exist. Decreasing: (-?, 1), (1, ?) 13) Increasing: (-4, ?) Decreasing: (-?, -4)

( ) ( ) 14) Increasing: (-?, 0),

2 , ?

2 Decreasing: 0,

3

3

( ) ( ) 15) Increasing: (-2, 0),

1 , ?

Decreasing: (-?, -2),

1 0,

2

2

16) Increasing: No intervals exist. Decreasing: (-?, -2), (-2, ?)

dy 5x2 17) =

dx 2 y

dy 9x2 18) = -

dx 10 y

dy

4

19) dx = 9 y2 + 2 y

dy 15x2y2 - 2x

21)

= dx

-3 - 10x3y

dy y3 - 6x 22) dx = -2 y - 3 y2x

dy -1 - 9x2y3

23) = dx

9x3y2

25) f ''(x) = -100x3 - 18x + 2 26) f '''(x) = 24x

27) f (4)(x) = 0

28) f '''(x) = 300x2 - 96x

dy 6x 20) =

dx 8 y + 1

dy -4 - 9xy3

24) = dx

9x2y2

dy (-2x2 + 5)-3 ? -2 - (-2x - 3) ? -3(-2x2 + 5)-4 ? -4x

29) =

dx

((-2x2 + ) 5)-3 2

= 2(-2x2 + 5)2(14x2 - 5 + 18x)

30) dy = (3x2 - 1)-3 ? -3x2 + (-x3 - 2) ? -3(3x2 - 1)-4 ? 6x

dx

3x(3x3 + x + 12)

=

(3x2 - 1)4

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Worksheet by Kuta Software LLC

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