To compare the estimated average rate of change of the functions \( b(x) = \sqrt[3]{3x + 9} \) and \( d(x) = \sqrt{-3x + 9} \) over the interval \([-12, -3]\), we need to understand the behavior of each function over that interval.
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Calculate the average rate of change for \( b(x) \):
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Evaluate \( b(-12) \) and \( b(-3) \):
- \( b(-12) = \sqrt[3]{3(-12) + 9} = \sqrt[3]{-36 + 9} = \sqrt[3]{-27} = -3 \)
- \( b(-3) = \sqrt[3]{3(-3) + 9} = \sqrt[3]{-9 + 9} = \sqrt[3]{0} = 0 \)
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Calculate the average rate of change: \[ \text{Average rate of change of } b(x) = \frac{b(-3) - b(-12)}{-3 - (-12)} = \frac{0 - (-3)}{9} = \frac{3}{9} = \frac{1}{3} \]
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Calculate the average rate of change for \( d(x) \):
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Evaluate \( d(-12) \) and \( d(-3) \):
- \( d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \)
- \( d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \)
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Calculate the average rate of change: \[ \text{Average rate of change of } d(x) = \frac{d(-3) - d(-12)}{-3 - (-12)} = \frac{3\sqrt{2} - 3\sqrt{5}}{9} \]
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Behavior of the functions in the interval:
- \( b(x) \) is an increasing function from \( b(-12) = -3 \) to \( b(-3) = 0 \).
- \( d(x) \) is decreasing since \( d(-12) > d(-3) \) (since \( \sqrt{45} > \sqrt{18} \)).
Based on this comparison:
- The average rate of change of \( b(x) \) is positive \((\frac{1}{3})\), while the average rate of change of \( d(x) \) is negative (because \( 3\sqrt{2} < 3\sqrt{5} \)).
The applicable statement from your options is: The estimated average rate of change of \( d(x) \) is greater than the estimated average rate of change of \( b(x) \), but both rates are negative.
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