Final answer:
To find the intervals on which the function (f) is increasing or decreasing, we calculate the derivative of f(x) and examine its sign. The increasing interval of f(x) is (π/4, 5π/4) and the decreasing interval is (5π/4, 9π/4).
Step-by-step explanation:
To find the intervals on which the function (f) is increasing or decreasing, we need to examine the derivative of the function. Let's start by finding the derivative of f(x):
f'(x) = 8 cos(x) - 8 sin(x)
Now, to determine where f'(x) is positive (increasing) or negative (decreasing), we need to find the values of x for which f'(x) is greater than zero or less than zero:
Increasing Interval: f'(x) > 0
8 cos(x) - 8 sin(x) > 0
cos(x) - sin(x) > 0
We can use the unit circle or trigonometric properties to find the solution, which is x ∈ (π/4, 5π/4).
Decreasing Interval: f'(x) < 0
8 cos(x) - 8 sin(x) < 0
cos(x) - sin(x) < 0
Again, using the unit circle or trigonometric properties, the solution is x ∈ (5π/4, 9π/4).