List The Intervals on Which F Is Decreasing Calculator

Determining where a function is decreasing is a fundamental calculus concept. This calculator helps you find the intervals on which a function f(x) is decreasing by analyzing its derivative. The result shows you exactly where the function slopes downward.

What Are Decreasing Intervals?

A function is decreasing on an interval if, as x increases, the value of f(x) decreases. Graphically, this means the function's curve is sloping downward on that interval. Mathematically, a function f(x) is decreasing on an interval (a, b) if for any two points x₁ and x₂ in (a, b) where x₁ < x₂, f(x₁) > f(x₂).

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

Next, solve the inequality f'(x) < 0. The solution to this inequality will give you the intervals where the function is decreasing.

Step 3: Consider Critical Points

Critical points occur where f'(x) = 0 or where f'(x) is undefined. These points divide the domain of the function into intervals that you should test separately.

Step 4: Test Each Interval

Choose a test point from each interval and plug it into f'(x). If the result is negative, the function is decreasing on that interval.

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
Factor: 3x(x - 2) < 0
Critical points: x = 0 and x = 2

Step 3: Test Intervals

Test x = -1: 3(-1)(-1 - 2) = 9 > 0 → Increasing
Test x = 1: 3(1)(1 - 2) = -3 < 0 → Decreasing
Test x = 3: 3(3)(3 - 2) = 9 > 0 → Increasing

The function is decreasing on the interval (0, 2).

Common Mistakes

When finding decreasing intervals, it's easy to make a few common errors:

Forgetting to consider critical points where the derivative is zero or undefined.
Miscounting the number of intervals to test.
Misapplying the test point method by choosing points that aren't in the interval.
Assuming the function is decreasing where the derivative is positive.

Double-checking your work and verifying with a graph can help avoid these mistakes.

FAQ

What does it mean if the derivative is zero?: The derivative being zero indicates a critical point where the function could be increasing, decreasing, or have a horizontal tangent. You need to test intervals around these points to determine the behavior.
Can a function be decreasing on multiple intervals?: Yes, a function can have multiple intervals where it is decreasing, especially if it has multiple peaks and valleys.
How do I know if my answer is correct?: You can verify your answer by testing points in each interval or by plotting the function to see where it slopes downward.
What if the derivative is undefined at a point?: Points where the derivative is undefined are also critical points. You should consider them when dividing the domain into intervals.