Intervals of Decreasing and Increasing Calculator

What are Intervals of Decreasing and Increasing?

In calculus, the intervals of decreasing and increasing for a function describe where the function's value either decreases or increases as the input variable changes. These intervals are determined by analyzing the first derivative of the function.

Why are these intervals important?

Understanding where a function increases or decreases helps in:

• Analyzing the behavior of functions
• Finding critical points and extrema
• Understanding the shape of graphs
• Solving optimization problems

How to Find Intervals of Decreasing and Increasing

To determine the intervals where a function is increasing or decreasing:

1. Find the first derivative of the function
2. Set the derivative equal to zero to find critical points
3. Determine the sign of the derivative in each interval between critical points
4. Identify where the derivative is positive (increasing) and negative (decreasing)

Steps in Detail

1. Differentiate the function to find f'(x).

2. Solve f'(x) = 0 to find critical points.

3. Create a number line with critical points and test intervals between them.

4. Determine the sign of f'(x) in each interval to classify as increasing or decreasing.

Worked Example

Let's find the intervals of increasing and decreasing for 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

• For x < 0: Test x = -1 → f'(-1) = 3(-1)² - 6(-1) = 9 > 0 → Increasing
• For 0 < x < 2: Test x = 1 → f'(1) = 3(1)² - 6(1) = -3 < 0 → Decreasing
• For x > 2: Test x = 3 → f'(3) = 3(3)² - 6(3) = 15 > 0 → Increasing

Result

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