To analyze the average rate of change of the function \( f(x) \) based on the given points (-1,0), (1,1), (3,0), (5,2), (6,2), and (7,0), we need to look at the slopes between key intervals.
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Interval \([1, 3]\):
- \( f(1) = 1 \)
- \( f(3) = 0 \)
- Average rate of change: \( \frac{f(3) - f(1)}{3 - 1} = \frac{0 - 1}{2} = -\frac{1}{2} \) (negative)
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Interval \([6, 7]\):
- \( f(6) = 2 \)
- \( f(7) = 0 \)
- Average rate of change: \( \frac{f(7) - f(6)}{7 - 6} = \frac{0 - 2}{1} = -2 \) (negative)
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Interval \([0, 1]\):
- \( f(0) \) is not given, but it appears to be between \( (-1, 0) \) and \( (1, 1) \), likely \( f(0) = 0 \) (assuming continuity or estimate).
- Average rate of change: \( \frac{f(1) - f(0)}{1 - 0} = \frac{1 - 0}{1} = 1 \) (not negative)
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Interval \([3, 4]\):
- \( f(3) = 0 \), but \( f(4) \) is not given; assuming it increases or remains at 0, we cannot conclude definitively about negativity.
Thus, if we analyze the intervals more critically:
- The average rate of change is negative in intervals \([1, 3]\) and \([6, 7]\).
Given this analysis, the response that best describes the graph of \( f(x) \) is:
- The average rate of change is negative on the interval [1,3] and also on the interval [6,7].
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