Identify the Open Interval on Which the Function is Increasing or Decreasing: cos(4x) – 3

To determine the interval on which the function f(x) = cos(4x) - 3 is increasing or decreasing, we first need to find its derivative.

The derivative of the function is calculated as follows:

f'(x) = d/dx[cos(4x) - 3] = -4sin(4x)

Now, we set the derivative equal to zero to find the critical points:

-4sin(4x) = 0

This occurs when sin(4x) = 0. The sine function is zero at integer multiples of π, hence:

4x = nπ, where n is any integer.

Therefore, we can solve for x:

x = nπ/4

Next, we analyze the intervals determined by these critical points. The values of n will give us points such as:

• x = -π/4, 0, π/4, π/2, ...

To determine if the function is increasing or decreasing, we can test the derivative f'(x) = -4sin(4x) in the intervals defined by these critical points:

• For n = -1, between -π/4 and 0:
• Pick a test point, say -π/8: f'(-π/8) = -4sin(-&pi/2) = 4 > 0 (increasing).
• For n = 0, between 0 and π/4:
• Pick a test point, say π/8: f'(π/8) = -4sin(&pi/2) = -4 < 0 (decreasing).
• For n = 1, between π/4 and π/2:
• Pick a test point, say 3π/8: f'(3π/8) = -4sin(3&pi/2) = 4 > 0 (increasing).