To determine where the function has a negative average rate of change, we need to look for intervals where the function's values decrease.
Let's evaluate the points within each provided interval:
-
Interval [0, 1]:
- At \( x = 0 \), \( y = -1 \)
- At \( x = 1 \), \( y = 0 \)
- The average rate of change: \[ \text{AROC} = \frac{0 - (-1)}{1 - 0} = \frac{1}{1} = 1 \] (Positive)
-
Interval [1, 2]:
- At \( x = 1 \), \( y = 0 \)
- At \( x = 2 \), \( y = -1 \)
- The average rate of change: \[ \text{AROC} = \frac{-1 - 0}{2 - 1} = \frac{-1}{1} = -1 \] (Negative)
-
Interval [2, 3]:
- At \( x = 2 \), \( y = -1 \)
- At \( x = 3 \), \( y = 1 \)
- The average rate of change: \[ \text{AROC} = \frac{1 - (-1)}{3 - 2} = \frac{2}{1} = 2 \] (Positive)
-
Interval [3, 5]:
- At \( x = 3 \), \( y = 1 \)
- At \( x = 5 \), \( y = 1 \)
- The average rate of change: \[ \text{AROC} = \frac{1 - 1}{5 - 3} = \frac{0}{2} = 0 \] (Neither positive nor negative)
From this analysis, we see that the only interval with a negative average rate of change is:
[1, 2].