On What Interval Is The Function Decreasing Calculator

A function is decreasing on an interval if its value decreases as the input increases. This calculator helps you determine where a function is decreasing by analyzing its derivative.

What Is a Decreasing Function?

A function f(x) is decreasing on an interval if for any two numbers x₁ and x₂ in that interval, where x₁ < x₂, the value of the function satisfies f(x₁) > f(x₂). In other words, as the input increases, the output decreases.

To determine where a function is decreasing, we typically analyze its derivative. If the derivative f'(x) is negative on an interval, then the function is decreasing on that interval.

How to Find Decreasing Intervals

Step 1: Find the Derivative

First, compute the derivative of the function f(x). This will give you f'(x), which represents the slope of the function at any point x.

Step 2: Determine Where the Derivative is Negative

Solve the inequality f'(x) < 0 to find the intervals where the derivative is negative. These are the intervals where the function is decreasing.

Step 3: Verify Critical Points

Check the critical points (where f'(x) = 0 or is undefined) to ensure you've identified all intervals where the function is decreasing.

Note: The function may be decreasing on multiple intervals. Always check the behavior between critical points.

Example Calculation

Let's find where the function f(x) = x³ - 3x² is decreasing.

Step 1: Find the Derivative

f'(x) = 3x² - 6x

Step 2: Solve f'(x) < 0

3x² - 6x < 0
x(x - 2) < 0

The solution to this inequality is 0 < x < 2. Therefore, the function is decreasing on the interval (0, 2).

FAQ

What if the derivative is zero on an interval?: The function is not decreasing where the derivative is zero. It's either constant or has a horizontal tangent.
Can a function be decreasing on multiple intervals?: Yes, a function can have multiple intervals where it's decreasing, especially if it has multiple critical points.
How do I know if the function is decreasing at a specific point?: Check the sign of the derivative at that point. If f'(x) < 0, the function is decreasing at x.