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On What Interval S Is F Increasing or Decreasing Calculator

Reviewed by Calculator Editorial Team

Determine where a function f is increasing or decreasing on interval s with our calculator and step-by-step guide. This tool helps you analyze the behavior of functions by identifying critical points and intervals of increase or decrease.

How to Use This Calculator

To determine where a function f is increasing or decreasing on interval s:

  1. Enter the function f(x) in the input field. Use standard mathematical notation (e.g., x^2, sin(x), e^x).
  2. Specify the interval s by entering the start and end values.
  3. Click "Calculate" to analyze the function.
  4. Review the results showing where the function is increasing or decreasing.

The calculator will display the intervals where the function is increasing or decreasing, along with critical points where the derivative is zero or undefined.

How It Works

This calculator determines where a function is increasing or decreasing by analyzing its derivative. A function is:

  • Increasing where its derivative is positive.
  • Decreasing where its derivative is negative.

The calculator follows these steps:

  1. Compute the derivative of the function f(x).
  2. Find critical points by solving f'(x) = 0 or where f'(x) is undefined.
  3. Test intervals between critical points to determine where the derivative is positive or negative.
  4. Return the intervals where the function is increasing or decreasing.
f'(x) = derivative of f(x) Critical points: f'(x) = 0 or f'(x) undefined Increasing: f'(x) > 0 Decreasing: f'(x) < 0

This method helps identify the behavior of the function on the specified interval.

Worked Example

Let's analyze the function f(x) = x^3 - 3x^2 on the interval s = [-1, 3].

  1. Compute the derivative: f'(x) = 3x^2 - 6x.
  2. Find critical points: 3x^2 - 6x = 0 → x(3x - 6) = 0 → x = 0 or x = 2.
  3. Test intervals:
    • For x < 0 (e.g., x = -1): f'(-1) = 3(-1)^2 - 6(-1) = 9 > 0 → Increasing.
    • For 0 < x < 2 (e.g., x = 1): f'(1) = 3(1)^2 - 6(1) = -3 < 0 → Decreasing.
    • For x > 2 (e.g., x = 3): f'(3) = 3(3)^2 - 6(3) = 15 > 0 → Increasing.

Results:

  • Increasing on [-1, 0] and [2, 3].
  • Decreasing on [0, 2].

FAQ

What is the difference between increasing and decreasing functions?
A function is increasing where its derivative is positive, meaning the function values are rising as x increases. It is decreasing where the derivative is negative, meaning the function values are falling as x increases.
How do I find critical points?
Critical points occur where the derivative is zero or undefined. Solve f'(x) = 0 or identify points where f'(x) is undefined.
Can this calculator handle piecewise functions?
Yes, you can enter piecewise functions using standard notation. The calculator will analyze each segment of the function separately.
What if the derivative is zero over an entire interval?
If the derivative is zero over an interval, the function is constant on that interval. The calculator will indicate this behavior.
How accurate are the results?
The calculator uses numerical methods to approximate derivatives and critical points. For precise results, ensure your function is correctly entered and the interval is appropriate.