Final answer:
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:
- Find the derivative of the function.
- Set the derivative equal to zero and solve for x to find the critical points.
- 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, ∞).