In two or more complete sentences, describe how to find the interval(s) where the function is decreasing and how interval notation is used to express the interval(s). In your final answer, include the interval in which the function is decreasing.
Question
In two or more complete sentences, describe how to find the interval(s) where the function is decreasing and how interval notation is used to express the interval(s). In your final answer, include the interval in which the function is decreasing.
Solution
To find the interval(s) where a function is decreasing, you need to analyze the sign of the derivative of the function.
- First, 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 by choosing test points within each interval and evaluating the sign of the derivative at those points.
- If the derivative is negative in an interval, it means the function is decreasing in that interval.
- Express the interval(s) where the function is decreasing using interval notation, which is written as (a, b) or [a, b], where a and b are the endpoints of the interval.
For example, if the derivative is negative between x = 2 and x = 5, the interval notation to express the interval where the function is decreasing would be (2, 5).
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