Final answer:
To find the interval on which the function is increasing, we take the derivative of the function and determine the intervals where the derivative is positive. In this case, the function f(x) = 5cos 2(x) - 10sin(x) is increasing on the interval (π/2, 3π/2) ∪ (3π/2, 2π).
Step-by-step explanation:
First, we need to find the derivative of the function f(x) = 5cos 2(x) - 10sin(x). Taking the derivative will help us determine the increasing or decreasing behavior of the function. The derivative of cos(x) is -sin(x) and the derivative of sin(x) is cos(x). Applying these derivative rules, we find that the derivative of f(x) is f'(x) = -10cos 2(x) - 10cos(x).
Next, we need to find the critical points by setting f'(x) = 0. Solving this equation, we get x = 0, x = π/2, x = 3π/2.
To determine the intervals on which f(x) is increasing, we use a test point within each interval and check if the derivative is positive. Plugging in a test point between x = 0 and x = π/2, such as x = π/4, into the derivative, we find that f'(π/4) = -10cos(2(π/4)) - 10cos(π/4) = -20√2 - 10√2 < 0. Therefore, f(x) is decreasing on the interval (0, π/2). Similarly, by plugging in test points for the other two intervals, we find that f(x) is increasing on the intervals (π/2, 3π/2) and (3π/2, 2π). Thus, the interval on which f(x) is increasing is (π/2, 3π/2) ∪ (3π/2, 2π).
Learn more about Increasing Intervals