compare the estimated average rate of change for the cubic function b\left(x\right)=\sqrt[3]{3x+9} to the estimated average rate of change of the square root function d\left(x\right)=\sqrt{-3x+9} over the interval [-12, -3]. which comparison is true?
A-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.
B-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.
C-the estimated average rate of change of b(x) is greater than the estimated average rate of change of d(x), but both rates are positive.
D-the estimated average rate of change of d(x) is greater than the estimated average rate of change of b(x) because d(x) is increasing over the interval but b(x) is decreasing.
To compare the average rate of change for the cubic function b(x) = (3x + 9)^(1/3) and the square root function d(x) = √(-3x + 9) over the interval [-12, -3], we can use the formula for average rate of change:
Average Rate of Change = (f(b) - f(a)) / (b - a)
For b(x) = (3x + 9)^(1/3), we have f(a) = b(-12) = (-27)^(1/3) = -3, and f(b) = b(-3) = 0. The average rate of change for b(x) is:
(b(x) - b(a)) / (b - a) = (0 - (-3)) / (-3 - (-12)) = 3/9 = 1/3
For d(x) = √(-3x + 9), we have f(a) = d(-12) = √(-33 + 9) = √(-24), and f(b) = d(-3) = √(-9 + 9) = 0. The average rate of change for d(x) is:
(d(x) - d(a)) / (b - a) = (0 - (√(-24))) / (-3 - (-12))
To simplify this expression, we can rewrite √(-24) as √(24)i, where i is the imaginary unit (√(-1)). We then have:
(d(x) - d(a)) / (b - a) = (0 - (√(24)i)) / (-3 + 12) = -√(24)i / 9
Comparing the average rates of change for both functions, we see that the average rate of change for b(x) is 1/3, which is greater than -√(24)i / 9. However, both rates are positive. Therefore, the correct comparison is:
C - The estimated average rate of change of b(x) is greater than the estimated average rate of change of d(x), but both rates are positive.