X 3 27x 5 Find The Interval of Increase Calculator

What is the Interval of Increase?

The interval of increase for a function is the set of x-values where the function's value increases as x increases. For a continuous function, this occurs where the derivative is positive.

For the function f(x) = x³ - 27x + 5, we'll find where f'(x) > 0 to determine the intervals of increase.

How to Find the Interval of Increase

Step 1: Find the First Derivative

First, calculate the derivative of the function f(x) = x³ - 27x + 5.

Step 2: Find Critical Points

Set the derivative equal to zero to find critical points where the function could change its increasing/decreasing behavior.

Step 3: Test Intervals

Test the intervals around the critical points to determine where the derivative is positive (function is increasing).

• For x < -3: Test x = -4 → f'(-4) = 3(16) - 27 = 48 - 27 = 21 > 0 → Increasing
• For -3 < x < 3: Test x = 0 → f'(0) = -27 < 0 → Decreasing
• For x > 3: Test x = 4 → f'(4) = 3(16) - 27 = 48 - 27 = 21 > 0 → Increasing

Example Calculation

Let's find the interval of increase for f(x) = x³ - 27x + 5.

1. Find the derivative: f'(x) = 3x² - 27
2. Find critical points: 3x² - 27 = 0 → x = ±3
3. Test intervals:
   • x < -3: f'(-4) = 21 > 0 → Increasing
   • -3 < x < 3: f'(0) = -27 < 0 → Decreasing
   • x > 3: f'(4) = 21 > 0 → Increasing

The function is increasing on the intervals (-∞, -3) and (3, ∞).

Interpretation of Results

The interval of increase shows where the function's value increases as x increases. For our example function:

• The function increases for all x-values less than -3
• The function decreases between -3 and 3
• The function increases again for all x-values greater than 3

This information is useful for understanding the behavior of the function and its turning points.