How to Find the Interval Where a Function is Increasing or Decreasing

To determine the intervals where a function is increasing or decreasing, you typically follow these steps:

1. Find the derivative: Start by taking the derivative of the function, which gives you the rate of change of the function.
2. Set the derivative to zero: Solve the equation where the derivative equals zero. This step helps you find critical points, where the function’s behavior might change.
3. Test intervals: Using the critical points, divide the number line into intervals. Choose a test point from each interval and plug it into the derivative.
4. Analyze the sign of the derivative: If the derivative is positive in an interval, the function is increasing there. If it is negative, the function is decreasing.

For example, consider the function f(x) = x^3 – 3x^2 + 4. First, we find the derivative:

f'(x) = 3x^2 - 6x

Next, set the derivative to zero:

3x^2 - 6x = 0 
=> 3x(x - 2) = 0 
=> x = 0, x = 2

Now, we test the intervals: (-∞, 0), (0, 2), and (2, ∞). Choosing test points (e.g., -1, 1, and 3):

• For x = -1: f'(-1) = 3(-1)^2 – 6(-1) = 3 + 6 = 9 (increasing)
• For x = 1: f'(1) = 3(1)^2 – 6(1) = 3 – 6 = -3 (decreasing)
• For x = 3: f'(3) = 3(3)^2 – 6(3) = 27 – 18 = 9 (increasing)

Therefore, the function is increasing on the intervals (-∞, 0) and (2, ∞), and decreasing on the interval (0, 2).