Final answer:
The function f(x) = e^(-x) - e^(-9x) has its absolute maximum at x=0 and absolute minimum at x=1 for the interval [0,1], which matches option B.
Step-by-step explanation:
To find the absolute minimum and maximum values of the function f(x) = e−x - e−9x on the interval [0,1], we first evaluate the function at the endpoints of the interval. At x = 0, f(x) simplifies to f(0) = e0 - e0 = 1 - 1 = 0. At x = 1, the function becomes f(1) = e−1 - e−9, which is a small positive number as e−1 is greater than e−9.
To find potential extrema within the interval, we calculate the derivative of f(x) and set it equal to zero, however the function is monotonic decreasing making it unnecesary to find critical points, moreover considering the provided information that the graph is a declining curve we can conclude the minimum will be at the right endpoint and the maximum at the left endpoint.
Therefore, the absolute maximum of f(x) occurs at x = 0, and the absolute minimum occurs at x = 1. The correct option in the final answer is B) Absolute minimum at x=1, absolute maximum at x=0.