5-4 Skills Practice: Analyzing Graphs of Polynomial Functions, with graphs and equations of polynomial functions and their key features.

Problem: Analyzing Graphs of Polynomial Functions

The task involves analyzing the behavior of polynomial functions at specific points and determining where relative minima and maxima occur. Below, I will solve each part step by step.

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#### 1. \( f(x) = x^3 - 3x \)

- Task: Determine the consecutive values of \( x \) between which each relative minimum and maximum occur.
- Given Table:
| \( x \) | \( f(x) \) |
|---------|------------|
| -4 | -56 |
| -3 | -18 |
| -2 | 2 |
| -1 | 2 |
| 0 | 0 |
| 1 | -2 |
| 2 | -2 |
| 3 | 18 |
| 4 | 56 |

- Analysis:
- A relative maximum occurs where the function changes from increasing to decreasing.
- A relative minimum occurs where the function changes from decreasing to increasing.
- From the table:
- Between \( x = -2 \) and \( x = -1 \): \( f(x) \) increases from 2 to 2 (no change).
- Between \( x = -1 \) and \( x = 0 \): \( f(x) \) decreases from 2 to 0.
- Between \( x = 0 \) and \( x = 1 \): \( f(x) \) decreases from 0 to -2.
- Between \( x = 1 \) and \( x = 2 \): \( f(x) \) remains constant at -2.
- Between \( x = 2 \) and \( x = 3 \): \( f(x) \) increases from -2 to 18.

- Relative Minimum: Occurs between \( x = -1 \) and \( x = 0 \), and between \( x = 1 \) and \( x = 2 \).
- Relative Maximum: Occurs between \( x = -2 \) and \( x = -1 \), and between \( x = 2 \) and \( x = 3 \).

- Solution:
- Relative minimum: \( (-1, 0) \) and \( (1, -2) \)
- Relative maximum: \( (-2, 2) \) and \( (2, -2) \)

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#### 2. \( g(x) = x^4 + 2x^2 + 2 \)

- Task: Determine the consecutive values of \( x \) between which each relative minimum and maximum occur.
- Given Table:
| \( x \) | \( g(x) \) |
|---------|------------|
| -3 | 95 |
| -2 | 22 |
| -1 | 5 |
| 0 | 2 |
| 1 | 5 |
| 2 | 22 |
| 3 | 95 |

- Analysis:
- The function \( g(x) \) is a polynomial of even degree with positive leading coefficient, so it opens upwards.
- From the table:
- Between \( x = -3 \) and \( x = -2 \): \( g(x) \) decreases from 95 to 22.
- Between \( x = -2 \) and \( x = -1 \): \( g(x) \) decreases from 22 to 5.
- Between \( x = -1 \) and \( x = 0 \): \( g(x) \) decreases from 5 to 2.
- Between \( x = 0 \) and \( x = 1 \): \( g(x) \) increases from 2 to 5.
- Between \( x = 1 \) and \( x = 2 \): \( g(x) \) increases from 5 to 22.
- Between \( x = 2 \) and \( x = 3 \): \( g(x) \) increases from 22 to 95.

- Relative Minimum: Occurs at \( x = 0 \) (since \( g(x) \) is symmetric and has a minimum value of 2 at \( x = 0 \)).
- Relative Maximum: None (the function does not have a relative maximum within the given interval).

- Solution:
- Relative minimum: \( (0, 2) \)
- Relative maximum: None

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#### 3. \( h(x) = 2x^3 + 3x^2 + 2x + 2 \)

- Task: Determine the consecutive values of \( x \) between which each relative minimum and maximum occur.
- Given Table:
| \( x \) | \( h(x) \) |
|---------|------------|
| -3 | -28 |
| -2 | -6 |
| -1 | 0 |
| 0 | 2 |
| 1 | 9 |
| 2 | 28 |

- Analysis:
- From the table:
- Between \( x = -3 \) and \( x = -2 \): \( h(x) \) increases from -28 to -6.
- Between \( x = -2 \) and \( x = -1 \): \( h(x) \) increases from -6 to 0.
- Between \( x = -1 \) and \( x = 0 \): \( h(x) \) increases from 0 to 2.
- Between \( x = 0 \) and \( x = 1 \): \( h(x) \) increases from 2 to 9.
- Between \( x = 1 \) and \( x = 2 \): \( h(x) \) increases from 9 to 28.

- The function is strictly increasing over the given interval, so there are no relative minima or maxima.

- Solution:
- Relative minimum: None
- Relative maximum: None

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#### 4. \( k(x) = 2x^4 + 2x^2 + 2 \)

- Task: Determine the consecutive values of \( x \) between which each relative minimum and maximum occur.
- Given Table:
| \( x \) | \( k(x) \) |
|---------|------------|
| -3 | 62 |
| -2 | 22 |
| -1 | 6 |
| 0 | 2 |
| 1 | 6 |
| 2 | 22 |
| 3 | 62 |

- Analysis:
- The function \( k(x) \) is a polynomial of even degree with positive leading coefficient, so it opens upwards.
- From the table:
- Between \( x = -3 \) and \( x = -2 \): \( k(x) \) decreases from 62 to 22.
- Between \( x = -2 \) and \( x = -1 \): \( k(x) \) decreases from 22 to 6.
- Between \( x = -1 \) and \( x = 0 \): \( k(x) \) decreases from 6 to 2.
- Between \( x = 0 \) and \( x = 1 \): \( k(x) \) increases from 2 to 6.
- Between \( x = 1 \) and \( x = 2 \): \( k(x) \) increases from 6 to 22.
- Between \( x = 2 \) and \( x = 3 \): \( k(x) \) increases from 22 to 62.

- Relative Minimum: Occurs at \( x = 0 \) (since \( k(x) \) is symmetric and has a minimum value of 2 at \( x = 0 \)).
- Relative Maximum: None (the function does not have a relative maximum within the given interval).

- Solution:
- Relative minimum: \( (0, 2) \)
- Relative maximum: None

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#### 5. \( m(x) = x^3 - 3x^2 - 3x + 2 \)

- Task: Determine the consecutive values of \( x \) between which each relative minimum and maximum occur.
- Given Table:
| \( x \) | \( m(x) \) |
|---------|------------|
| -3 | -55 |
| -2 | -18 |
| -1 | -3 |
| 0 | 2 |
| 1 | -3 |
| 2 | -8 |
| 3 | -7 |
| 4 | 18 |

- Analysis:
- From the table:
- Between \( x = -3 \) and \( x = -2 \): \( m(x) \) increases from -55 to -18.
- Between \( x = -2 \) and \( x = -1 \): \( m(x) \) increases from -18 to -3.
- Between \( x = -1 \) and \( x = 0 \): \( m(x) \) increases from -3 to 2.
- Between \( x = 0 \) and \( x = 1 \): \( m(x) \) decreases from 2 to -3.
- Between \( x = 1 \) and \( x = 2 \): \( m(x) \) decreases from -3 to -8.
- Between \( x = 2 \) and \( x = 3 \): \( m(x) \) increases from -8 to -7.
- Between \( x = 3 \) and \( x = 4 \): \( m(x) \) increases from -7 to 18.

- Relative Minimum: Occurs between \( x = -1 \) and \( x = 0 \), and between \( x = 1 \) and \( x = 2 \).
- Relative Maximum: Occurs between \( x = -2 \) and \( x = -1 \), and between \( x = 2 \) and \( x = 3 \).

- Solution:
- Relative minimum: \( (-1, -3) \) and \( (1, -3) \)
- Relative maximum: \( (-2, -18) \) and \( (2, -8) \)

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#### 6. \( n(x) = 3x^4 - 2x^3 - 2x^2 + 2 \)

- Task: Determine the consecutive values of \( x \) between which each relative minimum and maximum occur.
- Given Table:
| \( x \) | \( n(x) \) |
|---------|------------|
| -3 | 107 |
| -2 | 42 |
| -1 | 3 |
| 0 | 2 |
| 1 | -1 |
| 2 | 18 |
| 3 | 209 |

- Analysis:
- From the table:
- Between \( x = -3 \) and \( x = -2 \): \( n(x) \) decreases from 107 to 42.
- Between \( x = -2 \) and \( x = -1 \): \( n(x) \) decreases from 42 to 3.
- Between \( x = -1 \) and \( x = 0 \): \( n(x) \) increases from 3 to 2.
- Between \( x = 0 \) and \( x = 1 \): \( n(x) \) decreases from 2 to -1.
- Between \( x = 1 \) and \( x = 2 \): \( n(x) \) increases from -1 to 18.
- Between \( x = 2 \) and \( x = 3 \): \( n(x) \) increases from 18 to 209.

- Relative Minimum: Occurs between \( x = -1 \) and \( x = 0 \), and between \( x = 0 \) and \( x = 1 \).
- Relative Maximum: Occurs between \( x = -2 \) and \( x = -1 \), and between \( x = 1 \) and \( x = 2 \).

- Solution:
- Relative minimum: \( (-1, 3) \) and \( (0, 2) \)
- Relative maximum: \( (-2, 42) \) and \( (1, -1) \)

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Final Answers:

1. \( f(x) = x^3 - 3x \):
- Relative minimum: \( (-1, 2) \) and \( (1, -2) \)
- Relative maximum: \( (-2, 2) \) and \( (2, -2) \)

2. \( g(x) = x^4 + 2x^2 + 2 \):
- Relative minimum: \( (0, 2) \)
- Relative maximum: None

3. \( h(x) = 2x^3 + 3x^2 + 2x + 2 \):
- Relative minimum: None
- Relative maximum: None

4. \( k(x) = 2x^4 + 2x^2 + 2 \):
- Relative minimum: \( (0, 2) \)
- Relative maximum: None

5. \( m(x) = x^3 - 3x^2 - 3x + 2 \):
- Relative minimum: \( (-1, -3) \) and \( (1, -3) \)
- Relative maximum: \( (-2, -18) \) and \( (2, -8) \)

6. \( n(x) = 3x^4 - 2x^3 - 2x^2 + 2 \):
- Relative minimum: \( (-1, 3) \) and \( (0, 2) \)
- Relative maximum: \( (-2, 42) \) and \( (1, -1) \)

\[
\boxed{
\begin{aligned}
1. & \text{ Relative minimum: } (-1, 2), (1, -2); \text{ Relative maximum: } (-2, 2), (2, -2) \\
2. & \text{ Relative minimum: } (0, 2); \text{ Relative maximum: None} \\
3. & \text{ Relative minimum: None; Relative maximum: None} \\
4. & \text{ Relative minimum: } (0, 2); \text{ Relative maximum: None} \\
5. & \text{ Relative minimum: } (-1, -3), (1, -3); \text{ Relative maximum: } (-2, -18), (2, -8) \\
6. & \text{ Relative minimum: } (-1, 3), (0, 2); \text{ Relative maximum: } (-2, 42), (1, -1)
\end{aligned}
}
\]