To compare the estimated average rates of change for the functions \( b(x) = \sqrt[3]{3x + 9} \) and \( d(x) = \sqrt{-3x + 9} \) over the interval \([-12, -3]\), we need to first evaluate the functions at the endpoints of the interval.
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Evaluate \( b(x) \):
- \( b(-12) = \sqrt[3]{3(-12) + 9} = \sqrt[3]{-36 + 9} = \sqrt[3]{-27} = -3 \)
- \( b(-3) = \sqrt[3]{3(-3) + 9} = \sqrt[3]{-9 + 9} = \sqrt[3]{0} = 0 \)
Therefore, the average rate of change of \( b(x) \) from \(-12\) to \(-3\) is: \[ \text{Average rate of change of } b(x) = \frac{b(-3) - b(-12)}{-3 - (-12)} = \frac{0 - (-3)}{-3 + 12} = \frac{3}{9} = \frac{1}{3} \]
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Evaluate \( d(x) \):
- \( d(-12) = \sqrt{-3(-12) + 9} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \) (approximately 6.7)
- \( d(-3) = \sqrt{-3(-3) + 9} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \) (approximately 4.24)
Therefore, the average rate of change of \( d(x) \) from \(-12\) to \(-3\) is: \[ \text{Average rate of change of } d(x) = \frac{d(-3) - d(-12)}{-3 - (-12)} = \frac{3\sqrt{2} - 3\sqrt{5}}{-3 + 12} = \frac{3\sqrt{2} - 3\sqrt{5}}{9} \]
To compare the average rate of change:
- \( b(x) \) has an average rate of change of \( \frac{1}{3} \) (which is positive).
- \( d(x) \) subtracts a greater value from a smaller value, indicating the rate of change could be negative, depending on the roots of the square root.
After interpreting the average rates of change:
- \( b(x) \) is increasing in the interval, whereas \( d(x) \) holds a declining function \( \sqrt{-3x + 9} \).
From the provided multiple choices, the one that reflects the calculated information is:
The estimated average rate of change of \( b(x) \) is greater than the estimated average rate of change of \( d(x) \), because \( b(x) \) is increasing over the interval, but \( d(x) \) is decreasing.
Thus, the correct response is: The estimated average rate of change of \( b(x) \) is greater than the estimated average rate of change of \( d(x) \) because \( b(x) \) is increasing over the interval but \( d(x) \) is decreasing.