Final answer:
To determine the intervals where the function f(x) = x - 2sin(x) is increasing or decreasing, find the critical points by setting the derivative equal to zero or undefined. Test a point within each interval to determine if the function is increasing or decreasing. The correct option is A) (-∞, π/3) increasing, (π/3, 5π/3) decreasing, (5π/3, ∞).
Step-by-step explanation:
To determine the intervals where the function f(x) = x - 2sin(x) is increasing or decreasing, we need to find the critical points of the function. These points occur where the derivative of the function is equal to zero or undefined. Taking the derivative of f(x), we get f'(x) = 1 - 2cos(x).
Setting f'(x) equal to zero, we get 1 - 2cos(x) = 0. Solving for x, we find x = π/3 and x = 5π/3. These points divide the x-axis into three intervals: (-∞, π/3), (π/3, 5π/3), and (5π/3, ∞). We can now test a point within each interval to determine if the function is increasing or decreasing in that interval.
For example, if we test x = 0 in the first interval, we find that f'(0) = 1 - 2cos(0) = 1 - 2 = -1. Since f'(0) is negative, the function is decreasing in the interval (-∞, π/3). If we test x = π/2 in the second interval, we find that f'(π/2) = 1 - 2cos(π/2) = 1 - 0 = 1. Since f'(π/2) is positive, the function is increasing in the interval (π/3, 5π/3).
Finally, if we test x = 2π in the third interval, we find that f'(2π) = 1 - 2cos(2π) = 1 - 2 = -1. Since f'(2π) is negative, the function is decreasing in the interval (5π/3, ∞). Therefore, the correct option is A) (-∞, π/3) increasing, (π/3, 5π/3) decreasing, (5π/3, ∞)