Question

select all that apply
1 relative maximum at the point (-1,2)
2 relative minimum at the point (1,-2)
3 increasing on the interval (-1,1)
4 decreasing on the intervals (-∞, -1) and (1, ∞)
5 relative maximum at the point (1,-2)
6 relative minimum at the point (-1,2)
7 decreasing on the interval (-1,1)
8 increasing on the intervals (-∞, -1) and (1, ∞)

Explanation:

Step1: Analyze Relative Maximum/Minimum

A relative maximum is a point where the function changes from increasing to decreasing, and a relative minimum is where it changes from decreasing to increasing. If there's a relative maximum at \((-1, 2)\), the function increases before \(x = -1\) and decreases after. A relative minimum at \((1, -2)\) means it decreases before \(x = 1\) and increases after. So option 1 (relative max at \((-1,2)\)) and option 2 (relative min at \((1, -2)\)) are correct. Option 5 (relative max at \((1, -2)\)) and 6 (relative min at \((-1,2)\)) are incorrect.

Step2: Analyze Increasing/Decreasing Intervals

- For interval \((-1, 1)\): Between \(x = -1\) (relative max) and \(x = 1\) (relative min), the function should be decreasing (since it goes from max to min), so option 3 (increasing on \((-1,1)\)) is incorrect, option 7 (decreasing on \((-1,1)\)) would be correct (but let's check other intervals).
- For intervals \((-\infty, -1)\) and \((1, \infty)\): Before \(x = -1\) (relative max), the function should be increasing (to reach the max), and after \(x = 1\) (relative min), it should be increasing (after min). Wait, no—wait, if relative max at \((-1,2)\): when \(x < -1\), function is increasing (towards \((-1,2)\)), then decreasing from \(-1\) to \(1\) (towards \((1, -2)\)), then increasing after \(x = 1\). Wait, I made a mistake earlier. Let's re - correct:
- Relative max at \((-1,2)\): function increases on \((-\infty, -1)\) (to reach the max), then decreases on \((-1, 1)\) (to reach the min at \((1, -2)\)), then increases on \((1, \infty)\) (after the min). So:
- Decreasing intervals: \((-1, 1)\) (option 7 would be correct for decreasing on \((-1,1)\)), increasing intervals: \((-\infty, -1)\) and \((1, \infty)\) (option 8 would be correct for increasing on those intervals), option 4 (decreasing on \((-\infty, -1)\) and \((1, \infty)\)) is incorrect.

Wait, let's re - evaluate the options:
- Option 1: Relative maximum at \((-1,2)\): Correct, because at \(x=-1\), the function changes from increasing to decreasing.
- Option 2: Relative minimum at \((1, -2)\): Correct, because at \(x = 1\), the function changes from decreasing to increasing.
- Option 3: Increasing on \((-1,1)\): Incorrect, because between \(-1\) (max) and \(1\) (min), the function is decreasing.
- Option 4: Decreasing on \((-\infty, -1)\) and \((1, \infty)\): Incorrect, because on \((-\infty, -1)\) the function is increasing (towards max), and on \((1, \infty)\) it's increasing (after min).
- Option 5: Relative maximum at \((1, -2)\): Incorrect, \((1, -2)\) is a minimum.
- Option 6: Relative minimum at \((-1,2)\): Incorrect, \((-1,2)\) is a maximum.
- Option 7: Decreasing on \((-1,1)\): Correct (since from max at \(-1\) to min at \(1\), the function decreases).
- Option 8: Increasing on \((-\infty, -1)\) and \((1, \infty)\): Correct (increases to max at \(-1\), then increases after min at \(1\)).

Wait, but the original options: Let's check the given options again. The options are:
1. Relative maximum at \((-1,2)\)
2. Relative minimum at \((1, -2)\)
3. Increasing on \((-1,1)\)
4. Decreasing on \((-\infty, -1)\) and \((1, \infty)\)
5. Relative maximum at \((1, -2)\)
6. Relative minimum at \((-1,2)\)
7. Decreasing on \((-1,1)\)
8. Increasing on \((-\infty, -1)\) and \((1, \infty)\)

So correct options:
- Option 1: Correct (relative max at \((-1,2)\))
- Option 2: Correct (relative min at \((1, -2)\))
- Option 7: Correct (decreasing on \((-1,1)\))
- Option 8: Correct (increasing on \((-\infty, -1)\) and \((1, \infty)\))

Wait, but maybe the initial analysis of the graph (even though we don't have the graph, we can infer from relative max/min definitions)

Wait, let's start over with the definitions:

- **Relative Maximum**: A point \((a,f(a))\) is a relative maximum if \(f(a)\) is greater than the values of \(f(x)\) for \(x\) near \(a\). So at \(x=-1\), if \((-1,2)\) is a relative max, then for \(x\) close to \(-1\) (both sides), \(f(x)\leq f(-1) = 2\). So the function comes from below (increasing) to \((-1,2)\) and then goes down (decreasing) after \(x=-1\).

- **Relative Minimum**: A point \((b,f(b))\) is a relative minimum if \(f(b)\) is less than the values of \(f(x)\) for \(x\) near \(b\). So at \(x = 1\), if \((1, -2)\) is a relative min, then for \(x\) close to \(1\) (both sides), \(f(x)\geq f(1)=-2\). So the function comes from above (decreasing) to \((1, -2)\) and then goes up (increasing) after \(x = 1\).

Now, let's analyze each option:

1. **Relative maximum at \((-1,2)\)**: Correct, because the function changes from increasing (before \(x=-1\)) to decreasing (after \(x=-1\)).

2. **Relative minimum at \((1, -2)\)**: Correct, because the function changes from decreasing (before \(x = 1\)) to increasing (after \(x = 1\)).

3. **Increasing on \((-1,1)\)**: Incorrect, because between \(x=-1\) (max) and \(x = 1\) (min), the function is decreasing (from \(2\) to \(-2\)).

4. **Decreasing on \((-\infty, -1)\) and \((1, \infty)\)**: Incorrect, because on \((-\infty, -1)\) the function is increasing (towards the max at \((-1,2)\)) and on \((1, \infty)\) the function is increasing (after the min at \((1, -2)\)).

5. **Relative maximum at \((1, -2)\)**: Incorrect, \((1, -2)\) is a relative minimum (the function has a lower value here compared to nearby points).

6. **Relative minimum at \((-1,2)\)**: Incorrect, \((-1,2)\) is a relative maximum (the function has a higher value here compared to nearby points).

7. **Decreasing on \((-1,1)\)**: Correct, because between \(x=-1\) (max) and \(x = 1\) (min), the function's \(y\) - value decreases from \(2\) to \(-2\).

8. **Increasing on \((-\infty, -1)\) and \((1, \infty)\)**: Correct, because on \((-\infty, -1)\) the function is moving towards the maximum at \((-1,2)\) (so increasing) and on \((1, \infty)\) the function is moving away from the minimum at \((1, -2)\) (so increasing).

Answer:

1. Relative maximum at the point \((-1, 2)\), 2. Relative minimum at the point \((1, -2)\), 7. Decreasing on the interval \((-1, 1)\), 8. Increasing on the intervals \((-\infty, -1)\) and \((1, \infty)\)