To determine where the function \( f(x) \) is increasing based on the given points, we can evaluate the behavior of the graph within various intervals defined by the x-values where the function crosses the x-axis and has local extrema.
Given Points:
- Minimum at \( (-1.6, -56) \)
- Local Maximum at \( (0.8, 11.4) \)
- Minimum at \( (2, 0) \)
- Crossings at \( (-2.5, 0) \), \( (0, 0) \), and \( (2, 0) \)
Analysis of Intervals:
-
Interval \([-2.5, -1.6)\):
- As we move from \( -2.5 \) to \( -1.6 \), the function goes from \(0\) (at \( -2.5 \)) to \(-56\) (at \(-1.6\)), indicating that the function is decreasing.
-
Interval \([-2, -1]\):
- This interval contains the value \(0\) and would fall somewhere between \( -2.5\) and \(-1.6\). The behavior isn't directly known based on the provided data, but since \(f(x)\) is decreasing in the previous interval and likely remains so until reaching the minimum at \(-1.6\), there is not enough evidence to claim it is increasing here.
-
Interval \((-1.6, 0]\):
- In this interval, \(f(x)\) is moving from the minimum value at \( (-1.6, -56) \) to the crossing point at \( (0, 0) \). Hence, this portion of the function is increasing.
-
Interval \([0, 0.8)\):
- After crossing through the origin at \( (0, 0) \), and up to \( (0.8, 11.4) \), the function continues to increase. Thus, this interval shows that \( f(x) \) is increasing.
-
Interval \((0.8, 2)\):
- In this range, the function transitions from \( (0.8, 11.4) \) falling down to minimum at \( (2, 0) \), hence it is decreasing.
Conclusion:
Based on the analysis above, the intervals where \( f(x) \) is increasing are:
- \((-1.6, 0]\)
- \([0, 0.8)\)
So, you should select the options:
- \((-1.6, 0]\)
- \([0, 0.8)\)
1 answer