Question

How to find increasing and decreasing intervals of a function with fraction?

Answer 1

To find the increasing and decreasing intervals of a function with a fraction, you need to determine when the function is increasing or decreasing as the input values change.

Step-by-step explanation:

To find the increasing and decreasing intervals of a function with a fraction, you need to determine when the function is increasing or decreasing as the input values change.

Here's how you can do it:

1. Find the derivative of the function.
2. Set the derivative equal to zero and solve for x to find the critical points.
3. Test the intervals between the critical points using test points to see if the function is increasing or decreasing.

For example, let's say you have the function f(x) = (2x + 1)/(3x - 4). To find it's increasing and decreasing intervals, start by finding the derivative f'(x) = (6 - 2)/(3x - 4)^2. Setting the derivative equal to zero gives you the critical point x = 1.5. Now, test the intervals (-∞, 1.5) and (1.5, ∞) using test values like x = 0 and x = 2. If the function is positive in an interval, it is increasing. If it is negative, it is decreasing. In this case, the function is increasing on the interval (-∞, 1.5) and decreasing on the interval (1.5, ∞).

Answer 2

Increasing - if the graph goes up
Decreasing - if the graph is going down