Intervals of Increase and Decrease Calculus Calculator

Determine where a function increases or decreases using calculus. This calculator helps you find the intervals of increase and decrease for a given function by analyzing its first derivative.

What are intervals of increase and decrease?

In calculus, the intervals of increase and decrease describe the behavior of a function over its domain. A function is increasing where its derivative is positive, and decreasing where its derivative is negative.

Understanding these intervals helps analyze the function's growth and decay patterns, identify critical points, and determine maxima and minima.

How to find intervals of increase and decrease

Step 1: Find the first derivative

Start by finding the derivative of the function. This will give you the slope of the tangent line at any point.

Step 2: Determine critical points

Set the derivative equal to zero and solve for x to find critical points where the function might change its increasing or decreasing behavior.

Step 3: Test intervals around critical points

Divide the domain into intervals using the critical points. Test the sign of the derivative in each interval to determine if the function is increasing or decreasing.

Step 4: Analyze the results

Based on the sign of the derivative in each interval, conclude whether the function is increasing or decreasing.

To find intervals of increase and decrease: 1. Compute f'(x) 2. Find critical points where f'(x) = 0 3. Test intervals between critical points 4. Determine where f'(x) > 0 (increasing) or f'(x) < 0 (decreasing)

Worked example

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

Step 1: Find the first 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: Test intervals

Test the sign of f'(x) in the intervals (-∞, 0), (0, 2), and (2, ∞):

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

Step 4: Conclusion

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

FAQ

What does it mean when a function is increasing?

A function is increasing where its derivative is positive. This means the function's value increases as x increases.

What does it mean when a function is decreasing?

A function is decreasing where its derivative is negative. This means the function's value decreases as x increases.

Can a function be both increasing and decreasing?

Yes, a function can change between increasing and decreasing behavior at critical points where the derivative is zero.

What if the derivative is zero over an entire interval?

If the derivative is zero over an interval, the function is neither increasing nor decreasing on that interval.