Answer:
(B) one
Explanation:
You want to know how many points on the interval [0, 5] the function f(x) = e^(2x) have a slope equal to the average slope.
Rate of change
The instantaneous rate of change of function f(x) is its derivative:
f'(x) = 2e^(2x)
This is a continuously increasing function (as is f(x)), so in any given interval there will be only one point that has any given slope.
The Mean Value Theorem says there is at least one point in the interval with the same slope as the average slope. The nature of the derivative tells you there is exactly one point with the same slope as the average slope.
Where
The average rate of change on [0, 5] is ...
AROC = (e^(2·5) -e^(2·0))/(5 -0) = (e^10 -1)/5
The instantaneous rate of change will have that value where ...
f'(x) = 2e^(2x) = (e^10 -1)/5
2x = ln((e^10 -1)/10)
x = ln((e^10 -1)/10)/2 ≈ 3.84868475302
For this value of x, f'(x) = AROC