Final answer:
To find the average rate of change of a function over intervals, calculate the difference in values at the endpoints and divide it by the difference in x-values.
Step-by-step explanation:
To find the average rate of change of the function over the given intervals, we need to calculate the difference in the values of the function at the endpoints of the interval and divide it by the difference in the x-values of those endpoints.
For interval a) [5,7], the average rate of change can be found as follows:
f(7) - f(5) / (7 - 5)
Substituting the values into the equation, we have:
(12(7)^3 + 12) - (12(5)^3 + 12) / (7 - 5)
Simplifying the equation gives:
(12(343) + 12) - (12(125) + 12) / 2
Similarly, for interval b) [-4,4], the average rate of change can be found as:
f(4) - f(-4) / (4 - (-4))
Substituting the values and simplifying the equation gives:
(12(4)^3 + 12) - (12(-4)^3 + 12) / 8