List Intervals Where Function Is Increasing and Decreasing Calculator

Determine where a function is increasing or decreasing using our calculator. This tool helps you analyze the behavior of mathematical functions by identifying critical points and intervals.

How to Use This Calculator

To find where a function is increasing or decreasing:

Enter your function in the input field (e.g., "x^3 - 3x^2 + 4")
Specify the domain interval (e.g., -5 to 5)
Click "Calculate" to see the intervals where the function is increasing or decreasing

The calculator will display the intervals and show a graph of the function for visual confirmation.

Mathematical Methods

To determine where a function is increasing or decreasing, we use the first derivative test:

Find the first derivative of the function, f'(x)
Find the critical points by solving f'(x) = 0
Test the intervals between critical points to determine where f'(x) > 0 (increasing) or f'(x) < 0 (decreasing)

f'(x) = derivative of f(x) Critical points: f'(x) = 0 Increasing when: f'(x) > 0 Decreasing when: f'(x) < 0

This method works for continuous functions on closed intervals.

Worked Example

Let's analyze the function f(x) = x³ - 3x² + 4 on the interval [-2, 3].

Find the derivative: f'(x) = 3x² - 6x
Find critical points: 3x² - 6x = 0 → x(3x - 6) = 0 → x = 0 or x = 2
Test intervals:
- x < 0: f'(x) = 3(-1)² - 6(-1) = 9 > 0 → Increasing
- 0 < x < 2: f'(x) = 3(1)² - 6(1) = -3 < 0 → Decreasing
- x > 2: f'(x) = 3(3)² - 6(3) = 15 > 0 → Increasing

The function is increasing on [-2, 0] and [2, 3], and decreasing on [0, 2].

FAQ

What if the derivative is zero at a point?: The point is a critical point, and you need to test intervals around it to determine increasing or decreasing behavior.
Can this calculator handle piecewise functions?: Currently, the calculator works best with continuous functions. Piecewise functions may require manual analysis.
What if the function is not differentiable at a point?: Points where the function is not differentiable cannot be critical points. The calculator will skip these points in its analysis.