Intervals of Increase and Decrease of A Function Calculator

Determining the intervals where a function is increasing or decreasing is a fundamental calculus concept. This calculator helps you find these intervals by analyzing the derivative of a function.

What are Intervals of Increase and Decrease?

In calculus, the intervals of increase and decrease of a function describe where the function's value is rising or falling as the input variable changes. These intervals are determined by analyzing the first derivative of the function.

A function is increasing on an interval if its derivative is positive throughout that interval. Conversely, a function is decreasing on an interval if its derivative is negative throughout that interval.

Key Concept

Critical points (where the derivative is zero or undefined) divide the domain of a function into intervals. By testing the sign of the derivative in each interval, you can determine where the function is increasing or decreasing.

How to Find Intervals of Increase and Decrease

To find the intervals of increase and decrease for a function, follow these steps:

Find the derivative of the function, f'(x).
Determine the critical points by solving f'(x) = 0 or where f'(x) is undefined.
Plot the critical points on a number line to divide the domain into intervals.
Test the sign of f'(x) in each interval by choosing a test point from each interval.
Determine where f'(x) is positive (function increasing) and where it's negative (function decreasing).

Formula

If f'(x) > 0 on an interval, then f(x) is increasing on that interval.

If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.

Worked Example

Let's find the intervals of increase and decrease for the function f(x) = x³ - 3x².

Step 1: Find the derivative

f'(x) = 3x² - 6x

Step 2: Find critical points

Set f'(x) = 0: 3x² - 6x = 0 → 3x(x - 2) = 0 → x = 0 or x = 2

Step 3: Create intervals

The critical points divide the domain into three intervals: (-∞, 0), (0, 2), and (2, ∞).

Step 4: Test each interval

For (-∞, 0), test x = -1: f'(-1) = 3(-1)² - 6(-1) = 3 + 6 = 9 > 0 → Increasing
For (0, 2), test x = 1: f'(1) = 3(1)² - 6(1) = 3 - 6 = -3 < 0 → Decreasing
For (2, ∞), test x = 3: f'(3) = 3(3)² - 6(3) = 27 - 18 = 9 > 0 → Increasing

Result

The function f(x) = x³ - 3x² is increasing on (-∞, 0) and (2, ∞), and decreasing on (0, 2).

FAQ

What if the derivative is zero at a point?

A derivative of zero indicates a critical point. The function could be increasing, decreasing, or have a horizontal tangent at that point. You need to analyze the intervals around the critical point to determine the behavior.

Can a function change from increasing to decreasing without a critical point?

No, a function must have a critical point (where the derivative is zero or undefined) to change from increasing to decreasing or vice versa.

What if the derivative is undefined at a point?

An undefined derivative at a point is also a critical point. You should analyze the intervals around this point to determine the function's behavior.