Understanding Decreasing Intervals: Analyzing Math Functions

decreasing interval

where the graph has a negative slope (first derivative less than or equal to 0) inclusive interval

In mathematics, a decreasing interval is a specific range or interval of values on the number line where the function is decreasing.

To understand a decreasing interval, it is necessary to first understand what it means for a function to be decreasing. A function is considered decreasing if the values of the function decrease as the input values increase. In other words, as the independent variable increases, the dependent variable decreases.

In the context of a decreasing interval, we are interested in finding the range of values for the independent variable over which the function is decreasing. This interval consists of all the input values for which the function is decreasing.

To determine the decreasing interval of a function, you need to find the critical points and analyze the behavior of the function around these points. Critical points are points where the derivative of the function is either zero or undefined.

If the function is differentiable (meaning it has a derivative), you can find the critical points by setting the derivative equal to zero and solving the resulting equation. Once you have the critical points, you can use the first derivative test to determine if the function is decreasing or increasing around these points.

For example, let’s consider the function f(x) = x^2 – 4x + 3. To determine the decreasing interval, we first need to find the critical points. Taking the derivative of f(x), we get f'(x) = 2x – 4. Setting this derivative equal to zero, we get 2x – 4 = 0, which yields x = 2 as the critical point.

Now, we can use the first derivative test. For x < 2, the derivative is negative (e.g., f'(-1) = -6), indicating that the function is decreasing. For x > 2, the derivative is positive (e.g., f'(3) = 2), indicating that the function is increasing. Therefore, the decreasing interval for this function is (-∞, 2).

In summary, a decreasing interval refers to the range of values of the independent variable over which a function is decreasing. It is determined by finding the critical points of the function and analyzing the behavior of the function around these points using the first derivative test.

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