Answer :
Answer: (-∞, ∞), i.e.
Step-by-step explanation: To find the interval where the average rate of change is greatest, we need to find the interval where the derivative of f(x) is greatest.
First, we'll find the derivative of f(x):
f(x) = 5(2.6)^x
f'(x) = 5(2.6)^x * ln(2.6)
Now, we'll set the derivative equal to 0 and solve for x:
5(2.6)^x * ln(2.6) = 0
(2.6)^x = 0 (since ln(2.6) is not zero)
This is not possible, since 2.6 is not zero.
So, the derivative is never zero, and therefore the function is increasing over its entire domain. To find the interval where the average rate of change is greatest, we can look for the interval where the function is increasing at its fastest rate.
This happens when the derivative is largest, which occurs when x is large. In particular, as x approaches infinity, the derivative approaches infinity as well.
Therefore, the interval where the average rate of change is greatest is (-∞, ∞), i.e., the entire real line.
