xxf(x)f(x)

1

198

2

192

3

182

4

168

5

150

6

128

f(x)f(x) is Select an answer Increasing Decreasing  Select an answer Concave up Concave down

xxg(x)g(x)

1

128

2

150

3

168

4

182

5

192

6

198

g(x)g(x) is Select an answer Increasing Decreasing  Select an answer Concave up Concave down

xxh(x)h(x)

1

2

2

8

3

18

4

32

5

50

6

72

h(x)h(x) is Select an answer Increasing Decreasing  Select an answer Concave up Concave down

xxk(x)k(x)

1

72

2

50

3

32

4

18

5

8

6

2

k(x)k(x) is Select an answer Increasing Decreasing  Select an answer Concave up Concave down

Let's analyze each function \( f(x) \), \( g(x) \), \( h(x) \), and \( k(x) \) based on the given values to determine whether they are increasing or decreasing, and whether they are concave up or concave down.

### 1. Analyzing \( f(x) \)
- Values for \( f(x) \):
  - \( f(1) = 198 \)
  - \( f(2) = 192 \)
  - \( f(3) = 182 \)
  - \( f(4) = 168 \)
  - \( f(5) = 150 \)
  - \( f(6) = 128 \)

#### Increasing/Decreasing:
The values of \( f(x) \) are getting smaller as \( x \) increases, so \( f(x) \) is **decreasing**.

#### Concavity:
Let's look at the second differences (changes in the changes of \( f(x) \)):

- \( 192 - 198 = -6 \)
- \( 182 - 192 = -10 \)
- \( 168 - 182 = -14 \)
- \( 150 - 168 = -18 \)
- \( 128 - 150 = -22 \)

Since the differences are becoming larger in the negative direction, the rate of decrease is increasing. This indicates that \( f(x) \) is **concave down**.

### 2. Analyzing \( g(x) \)
- Values for \( g(x) \):
  - \( g(1) = 128 \)
  - \( g(2) = 150 \)
  - \( g(3) = 168 \)
  - \( g(4) = 182 \)
  - \( g(5) = 192 \)
  - \( g(6) = 198 \)

#### Increasing/Decreasing:
The values of \( g(x) \) are getting larger as \( x \) increases, so \( g(x) \) is **increasing**.

#### Concavity:
Let's look at the second differences:

- \( 150 - 128 = 22 \)
- \( 168 - 150 = 18 \)
- \( 182 - 168 = 14 \)
- \( 192 - 182 = 10 \)
- \( 198 - 192 = 6 \)

Since the differences are becoming smaller, the rate of increase is decreasing. This indicates that \( g(x) \) is **concave down**.

### 3. Analyzing \( h(x) \)
- Values for \( h(x) \):
  - \( h(1) = 2 \)
  - \( h(2) = 8 \)
  - \( h(3) = 18 \)
  - \( h(4) = 32 \)
  - \( h(5) = 50 \)
  - \( h(6) = 72 \)

#### Increasing/Decreasing:
The values of \( h(x) \) are getting larger as \( x \) increases, so \( h(x) \) is **increasing**.

#### Concavity:
Let's look at the second differences:

- \( 8 - 2 = 6 \)
- \( 18 - 8 = 10 \)
- \( 32 - 18 = 14 \)
- \( 50 - 32 = 18 \)
- \( 72 - 50 = 22 \)

Since the differences are becoming larger, the rate of increase is increasing. This indicates that \( h(x) \) is **concave up**.

### 4. Analyzing \( k(x) \)
- Values for \( k(x) \):
  - \( k(1) = 72 \)
  - \( k(2) = 50 \)
  - \( k(3) = 32 \)
  - \( k(4) = 18 \)
  - \( k(5) = 8 \)
  - \( k(6) = 2 \)

#### Increasing/Decreasing:
The values of \( k(x) \) are getting smaller as \( x \) increases, so \( k(x) \) is **decreasing**.

#### Concavity:
Let's look at the second differences:

- \( 50 - 72 = -22 \)
- \( 32 - 50 = -18 \)
- \( 18 - 32 = -14 \)
- \( 8 - 18 = -10 \)
- \( 2 - 8 = -6 \)

Since the differences are becoming smaller in magnitude, the rate of decrease is decreasing. This indicates that \( k(x) \) is **concave up**.

### Summary:
1. \( f(x) \) is **decreasing** and **concave down**.
2. \( g(x) \) is **increasing** and **concave down**.
3. \( h(x) \) is **increasing** and **concave up**.
4. \( k(x) \) is **decreasing** and **concave up**.

Would you like any further details or explanations?

Here are 5 related questions you could explore:
1. What are the properties of functions that are concave up or concave down?
2. How do second derivatives relate to concavity in functions?
3. Can a function change from concave up to concave down, and how can we identify this point?
4. What is the significance of decreasing and increasing intervals in real-world applications?
5. How can concavity and monotonicity help in optimizing functions?

**Tip**: To determine concavity, check how the rate of change (first differences) is evolving—if it accelerates, the function is concave up; if it decelerates, it’s concave down.

The query submitted by the user: f(x)f(x)

1

198

2

192

3

182

4

168

5

150

6

128

f(x)f(x) is Select an answer Increasing Decreasing Select an answer Concave up Concave down

g(x)g(x)

1

128

2

150

3

168

4

182

5

192

6

198

g(x)g(x) is Select an answer Increasing Decreasing Select an answer Concave up Concave down

h(x)h(x)

1

2

2

8

3

18

4

32

5

50

6

72

h(x)h(x) is Select an answer Increasing Decreasing Select an answer Concave up Concave down

k(x)k(x)

1

72

2

50

3

32

4

18

5

8

6

2

k(x)k(x) is Select an answer Increasing Decreasing Select an answer Concave up Concave down

This problem involves analyzing the monotonicity (increasing or decreasing behavior) and concavity (concave up or down) of four functions: f(x), g(x), h(x), and k(x). Using first and second differences, we determine that f(x) and k(x) are decreasing and concave down/up, while g(x) and h(x) are increasing and concave down/up. Suitable for students in Grades 10-12.

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