Answer :
To determine the intervals on which a function is increasing or decreasing, we generally look at the first derivative of the function. The sign of the first derivative indicates the behavior of the function:
- If the first derivative is positive, the function is increasing.
- If the first derivative is negative, the function is decreasing.
- If the first derivative is zero at certain points, those points are critical points and can indicate local maxima, minima, or points of inflection.
Let's analyze the given results:
### Intervals on Which the Function is Increasing
Given the correct answer:
- The function is increasing on the intervals [tex]\((-\infty, -7)\)[/tex] and [tex]\((-2, -1)\)[/tex].
So, in interval notation, the function is increasing on:
[tex]$ (-\infty, -7), (-2, -1) $[/tex]
### Intervals on Which the Function is Decreasing
Given that there are no intervals listed for decreasing intervals, the function is not decreasing on any interval. Therefore, in interval notation:
[tex]$ \text{There are no intervals on which the function is decreasing.} $[/tex]
### Intervals on Which the Function is Constant
No information about the function being constant is provided, so we can assume that the function does not remain constant on any interval.
### Summary
- The function is increasing on the intervals [tex]\((-\infty, -7)\)[/tex] and [tex]\((-2, -1)\)[/tex].
- The function is decreasing on no intervals (empty set).
Therefore, listing the intervals where the function is increasing and decreasing:
- Increasing: [tex]\((-\infty, -7), (-2, -1)\)[/tex]
- Decreasing: No intervals (empty set)
- If the first derivative is positive, the function is increasing.
- If the first derivative is negative, the function is decreasing.
- If the first derivative is zero at certain points, those points are critical points and can indicate local maxima, minima, or points of inflection.
Let's analyze the given results:
### Intervals on Which the Function is Increasing
Given the correct answer:
- The function is increasing on the intervals [tex]\((-\infty, -7)\)[/tex] and [tex]\((-2, -1)\)[/tex].
So, in interval notation, the function is increasing on:
[tex]$ (-\infty, -7), (-2, -1) $[/tex]
### Intervals on Which the Function is Decreasing
Given that there are no intervals listed for decreasing intervals, the function is not decreasing on any interval. Therefore, in interval notation:
[tex]$ \text{There are no intervals on which the function is decreasing.} $[/tex]
### Intervals on Which the Function is Constant
No information about the function being constant is provided, so we can assume that the function does not remain constant on any interval.
### Summary
- The function is increasing on the intervals [tex]\((-\infty, -7)\)[/tex] and [tex]\((-2, -1)\)[/tex].
- The function is decreasing on no intervals (empty set).
Therefore, listing the intervals where the function is increasing and decreasing:
- Increasing: [tex]\((-\infty, -7), (-2, -1)\)[/tex]
- Decreasing: No intervals (empty set)
